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Note: We have an additional 30 sample questions for this chapter not included in this printable section.
Number Rules
Consecutive Numbers
Divisibility
Fractions
Decimals
Exponents
Roots & Radicals
Note: this section is designed as an introduction for students scoring in the middle to low ranges.
Number Definitions
Integer |
a member of the set of positive whole numbers {1, 2, 3, . . . }, negative whole numbers {-1, -2, -3, . . . }, and zero. Fractions and decimals are not integers. Integers are also called "whole numbers". |
Positive number |
a number greater than zero, such as +5 (usually written simply as 5). |
Negative number |
a number less than zero, such as -5. |
Zero |
is neither positive nor negative and is an integer. |
Rational number |
a number that may be expressed as integers, decimals or fractions (a ratio of integers), as opposed to an... |
Irrational number |
a number that cannot be expressed by the ratio of two integers, such as π or √2. Very mysterious!
Note: anything divided by zero is irrational. |
If you forget a rule (this happens all the time under test day pressure) consider using Experiments or Backsolving/Plug-In to get around it (we discuss these advanced strategies in II. General Math Strategies.
ORDER OF OPERATIONS - PEMDA
| Take a look at: |
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| 5 + 2 × 3, what is it? |
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| (5 + 2) × 3 = (7) × 3 = 21 |
Add first, then multiply |
or? |
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| 5 + (2 × 3 ) = 5 + (6) = 11 |
Multiply first, then add |
To address this issue we have the "Order of Operations," which is the priority list for calculations. Parentheses go first, and then followed by exponents. Multiplication and division are next, but are interchangeable depending on which comes up first when going from left to right in a problem. The same is true for addition and subtraction. There is a handy mnemonic to memorize these rules.
Rule I: Please Excuse My Dear Aunt Sally
| Please |
Parentheses |
(-5 + 2)(-3) = (-3)(-3) = 9 |
| Excuse |
Exponents |
22 + 4 = 4 + 4 = 8 |
| My |
Multiplication |
5 + (2 × 3) = 5 + (6) = 11 |
| Dear |
Division |
8 ÷ 2 + 2 = 4 + 2 = 6 |
| Aunt |
Addition |
4 + 6 - 10 = 10 - 10 = 0 |
| Sally |
Subtraction |
7 - 5 - 2 = 2 - 2 = 0 |
| Rule II: Combine all like terms in an expression. |
2x + x - y + 4y
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3x + 3y |
The expression is simplified by combining the like terms (x's and y's). |
| Rule III: Distribute numbers to eliminate parenthesis. This means multiply everything within the parenthesis. |
2(x - y) = 2x - 2y |
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-2(x - y) = -2x + 2y |
Multiply the negative through to make a positive number. |
Rule IV: Eliminate inner parentheses first and the outermost parentheses last.
In the expression: |
x(x + 2(3x + 4) -3) |
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x(x + 6x + 8 - 3) |
distribute the inner parentheses: 2(3x + 4) = 6x + 8 |
x(7x + 5) |
combine like terms (both integers and the x's) inside the parenthesis. |
7x2 + 5x
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remove the last parentheses through distribution. |
Simplify 16 ÷ 2(8 - 3(4 - 2)) + 3
Solution
16 ÷ 2(8 - 3(4 - 2)) + 3 |
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16 ÷ 2(8 - 3(2)) + 3 |
combine the inner parentheses: (4 - 2) = 2 |
16 ÷ 2(2) + 3 |
combine the inner parentheses: 8 - 3(2) = 2 |
16 ÷ 4 + 3
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multiply out the 2(2) = 4 |
4 + 3 = 7
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16 ÷ 4 = 4. Answer = 7 |
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Simplify 16 - 3(8 - 3)2 ÷ 15
Solution
16 - 3(8 - 3)2 ÷ 15 |
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16 - 3(5)2 ÷ 15 |
combine the inner parentheses: (8 - 3) = 5 |
16 - 3(25) ÷ 15 |
multiply out the exponent: (5)2 = 25 |
16 - 75 ÷ 15
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multiply out 3(25) = 75 |
16 - 5 = 11
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75 ÷ 15 = 5. Answer = 11 |
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Odd / Even Rules
Even number |
an integer that is divisible by 2 (0, 2, 4, 6, 8, 10).
No, don't email us saying that's a typo, zero is an even integer. |
Odd number |
an integer not divisible by 2 (3, 5, 7, 9). |
Prime number |
a positive integer with exactly two different positive divisors: 1 and itself. For example, 2, 3, 5, 7 are prime numbers.
Note: 1 is not considered a prime number since it has only one positive divisor. |
Adding, Subtracting and Multiplying Odd/Even: The following is true of even and odd whole numbers (use example numbers in your mind to illustrate).
Rule |
Example |
Even + Even = Even |
4 + 4 = 8 (even) |
Odd + Even = Odd |
3 + 4 = 7(odd) |
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Odd + Odd = Even
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3 + 3 = 6 (even) |
Even × Even = Even
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2 × 2 = 4 (even) |
| Even × Odd = Even |
2 × 3 = 6 (even) |
| Odd × Odd = Odd |
3 × 3 = 9 (odd) |
| Even - Even = Even |
16 - 8 = 8 (even) |
| Even - Odd = Odd |
16 - 5 = 11 (odd) |
| Odd - Odd = Even |
9 - 5 = 4 (even) |
| Examples (with k as an odd number) |
| a) k + k + k |
(k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd. |
| b) k × k × k |
k × k is odd. Thus (k × k) × k is an odd times an odd, which is odd. |
| c) k + 2k |
k + 2k is an odd plus an even, which is odd. |
| d) 2k × k |
2k is even. An even times an odd is even. |
Positive/Negative Rules
| Adding a negative number is the same as subtraction. |
4 + (-5) = 4 - 5 = -1 |
Subtracting a negative number is the same as addition.
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4 - (-5) = 4 + 5 = 9
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| ALWAYS REMEMBER: DOUBLE NEGATIVE = POSITIVE. If you see two negative signs you can cancel them out and make a plus sign. |
3 + 4 - (4 - 6) |
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3 + 4 - (-2) |
Combine numbers in parenthesis (4 - 6). |
3 + 4 + 2 = 9 |
double negative = positive: - (-2) = + 2. |
Multiplication/Division
When multiplying or dividing: if they have the same sign, then it is POSITIVE. If they have different signs, then it is NEGATIVE.
| Positive × Positive = Positive |
2 × 2 = 4 |
| Positive × Negative = Negative |
2 × -2 = -4 |
| Negative × Negative = Positive |
-2 × -2 = 4 |
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800score Strategies
Use Plug In or Backsolving on odd/even and positive/negative questions.
Make up numbers to substitute in for variables on questions that ask if variables result in a positive or negative number. For example, if x < 0, is x2 positive? Sub in -2 for x and you can see that (-2)2 is positive. This is also very useful for double checking.
Jot down all possibilities when dealing with discrete sets
If a positive, negative or prime number question asks "How many two digit prime numbers" or "How many two digit numbers less than 50" then you have a finite (small) number of possibilities. Here it is useful to sometimes make a list of all the number possibilities rather than try to solve the question using algebra.
If you forget an odd/even addition/subtraction rule, use Experiments to derive it in the middle of the test.
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Absolute Value
The absolute value of a number, written as | -5 |, is the distance of a number from zero on the number line:
| +5 | = | -5 | = 5. This means that absolute value questions usually have two possible answers (one positive and one negative).
|x + 5| = 1 |
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x + 5 = 1
x + 5 = -1 |
To get rid of the absolute value symbol, create negative (-1) and positive (+1) scenarios. |
x = -4
x = -6 |
Two solutions: x = -4/-6 |
Powers
Writing a number raised to a certain power is a shorthanded way of expressing multiplication of a number by itself. For example:
52 = 5 × 5 = 25
43 = 4 × 4 × 4 = 64
In these two cases, 5 and 4 are the respective bases, and 2 and 3 are the powers, or the number of times we multiply a number by itself.
Consecutive numbers are a set of numbers in which each member of the set is the successor of its predecessor.
| 4, 6, 8, 10 |
even consecutive numbers |
| 3, 5, 7, 9 |
odd consecutive numbers |
| 3, 5, 7, 11, 13 |
prime consecutive numbers: |
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Beware of "inclusive questions"
When the GMAT says "inclusive" it means that they are counting the first and the last of a given set. Sometimes the GMAT doesn't even say "inclusive" and it is implied in the question.
How many numbers are there from 2 to 6?
Normally you think to subtract. 6 - 2 = 4. But, try counting the numbers from 2 to 6. 2, 3, 4, 5, 6. The answer is 5. |
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A fence consists of a fence post (1 foot wide) and a bridge connecting each post that is 9 feet long. How many fence posts would be required for a 100 foot fence?
| Solution
The trick here is to realize that at the end you have an extra post. This means that there would be a total of 10 posts, plus the final one (11 posts).
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Sum of Consecutive Integers
What is the sum of the numbers from 1 to 100?
Now, if you are taking the GMAT and scribbling on your dry erase board numbers from 1 to 100, it means that you are wasting your time! Surely the GMAT would never have you do such labor intensive mathematics. So how could you solve such an onerous problem?
How to solve these questions:
In general to find the sum of all the numbers from F to L (where F is the first number and L is the last number), use the formula:
(F + 1) + (F + 2) + (F + 3) + (F + 4) + . . . . + L = (F + L) × (L - F + 1) / 2
(F + L) × (L/2) |
F = 1, L = 100 |
(1 + 100) × (100/2) |
Plug in values |
101 × 50 = 5050 |
50
Sum of numbers from 1 to 100 is 5050 |
Consecutive Number Magic Tricks
(NOTE: ADVANCED CONTENT - lower scorers consider skipping this section and coming back to it later).
- Consecutive Number Magic Trick #1
If you have an ODD (not even) number of numbers in your set, the sum will always be a multiple of the number of numbers.
5 numbers in this set of consecutive numbers:
1 + 2 + 3 + 4 + 5 = 15.
YES, 15 is a multiple of 5.
6 numbers in a set of consecutive numbers:
11 + 12 + 13 + 14 + 15 + 16 = 81.
NO, and there are 6 numbers (an EVEN number of numbers). 81 isn't a multiple of 6.
What is the secret to this trick?
Let's try out the algebra with a set of 5 consecutive integers (an ODD number of numbers).
(x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) = 5x + 15
5x + 15 is a multiple of 5
- Consecutive Number Magic Trick #2
If you multiply an ODD number of numbers in a consecutive number set, it WILL ALWAYS be divisible by the number of numbers.
For example, a series of 5 consecutive integers, it WILL ALWAYS be divisible by 5.
2 × 3 × 4 × 5 × 6 = 720
Here there are 5 numbers. Why does this rule work? Because a group of 5 consecutive numbers MUST include a multiple 5 in it. (2, 3, 4, 5, 6) or (18, 19, 20, 21, 22).
Try this example:
6 numbers in a row. Here we added a 1 in the front.
1 × 2 × 3 × 4 × 5 × 6 = 720
Here there are 6 numbers, and yes, 720 is divisible by 6.
- Number Series Number Magic Trick #3
In the PRODUCT of a number series: if it has one even number (multiple of 2), then the product is a multiple of 2.
6 × 7 = 42
If it has TWO even numbers, then it is a multiple of 4, ALWAYS:
6 × 7 × 8 = 336.
Abracadabra! 336 is a multiple of 4 (4 × 84)
If it has THREE even numbers, then it is a multiple of 8, ALWAYS:
6 × 7 × 8 × 9 × 10 = 30240
Abracadabra! 30240 is a multiple of 8 (3780 × 8 = 30240)
Ok, 800score, explain this one! If the product of several numbers contains only ONE even number, the result will be even because anything multiplied by an even number is ALWAYS even. So that one is easy.
The second one: a product of a consecutive number series that contains TWO even numbers must be a multiple of 4. You can figure this one out by doing algebra.
Here is our consecutive numbers series:
x × (x + 1) × (x + 2)
If x is even, then (x + 2) must be even as well. Therefore, you have two even numbers as factors. If you prime factorize the number you must have two 2's in it. Therefore, the product of 2 × 2 = 4 MUST be a divisor (factor) of the number series.
Head spinning? Plug-In numbers:
4 × 5 × 6
6 has 2 as a factor and 4 has 2 as a factor, therefore your end result product must be a multiple of 2 × 2 = 4.
Go one step further: if you have 3 even numbers in a consecutive number series, the product of that series must have three 2's as prime factors. Therefore, the result must have 8 as a factor.
Divisibility
Prime Numbers
Greatest Common Factor
Least Common Multiple
DIVISIBILITY
| Number |
Rule of divisibility |
1 |
All whole numbers are divisible by 1. |
2 |
All non-zero numbers with a ones digit of 0, 2, 4, 6, or 8 are divisible by 2. |
3 |
If the sum of a number's digits are divisible by 3, then it is divisible by 3. For example, 222 has digits add to 6, so the whole number is divisible by 3 (222/3= 74). |
4 |
A number is divisible by 4 if its last two digits are divisible by 4. For example, 60,548 is divisible by 4, but 6,250 is not. |
5 |
A non-zero number is divisible by 5 if it ends in 0 or 5. |
6 |
A number is divisible by 6 if it is even (divisible by 2) and also divisible by 3. Any even number divisible by 3 is also divisible by 6. |
9 |
Add the digits. If that sum is divisible by nine, then the original number is as well.
For example, 1,044 (1044/9 =116) is divisible by 9 (because 1 + 0 + 4 + 4 = 9. |
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10 |
A non-zero number is divisible by 10 if it ends in 0. |
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Sums, differences and products of factors are factors
10 is a factor of 100
10 is a factor of 50
100 + 50 = 150 (a factor of 10)
100 - 50 = 50 (a factor of 10)
100 × 50 = 5000 (a factor of 10)
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Factors (OR Divisors)
A factor (also called a divisor) is an integer that divides another number resulting in a whole number. This is expressed by the rule:
- If a/b is an integer, then b is a factor of a.
For example, 4 is a factor of 20 because 20/4 = 5. On the other hand, 8 is not a factor of 20 because 20/8 is 2.5, and 2.5 is not an integer.
To factorize, just break down a number into as many factors as possible. For example, the factors of 12 are 1, 2, 3, 4 and 6.
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
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Don't forget!
When a question asks... "how many factors does... have?"
Always remember to include:
- 1, because 1 is a factor of ALL integers itself.
- The number itself. For example, 21 is a factor of 21.
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State all the factors of 63.
| Solution
The number 63 can be divided by 1, 3, 7, 9, 21, and 63. Hence, these are its factors.
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Prime Numbers
An integer greater than one is called a prime number if its only positive factors (divisors) are 1 and the number itself.
Sample prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (these common primes should be memorized).
Note: 1 is not a prime number and, aside from 2, all prime numbers are ODD numbers. This means that the sum of two primes must be even UNLESS one of the primes is 2.
Prime Factorization
Using your knowledge of primes, you can prime factorize a number. Sounds like fun? When you prime factorize, you break a number into its prime factors.
So 36 is broken into:
36 = 12 × 3
36 = (2 × 2 × 3) × 3
| Row |
Prime Factorization Chart |
| 1 |
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36 |
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| 2 |
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12 |
3 |
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2 |
2 |
3 |
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Here the red numbers are all prime factors.
Note: all the factors in row 2 and 3 are factors of 36. Any number in the prime factorization chart is a factor of any number above it. THIS IS A POWERFUL TOOL ON THE GMAT. Using this tool can tell if any number is a factor of another number,
Is 36 a factor of 81? |
Prime Factorization Chart |
36 |
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12 |
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3 |
2 |
2 |
3 |
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36 breaks into prime factors 2, 2, 3, 3 |
Prime Factorization Chart |
81 |
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27 |
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3 |
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9 |
3 |
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3 |
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81 breaks into 3, 3, 3, 3 |
You can't make 36 from any combination of 3, 3, 3, 3 (the factors of 81) because 36 has (2, 2) in it and 81 does not. So 36 is NOT a factor of 81. BUT, if you could put (2, 2) into 81, the result is 324 (81 × 4), and then 36 then becomes a factor of 324 (9 × 36 = 324).
If the integer n is divisible by 3, 5 and 13, what other numbers must be divisors of n?
Solution
Since we know that 3, 5, and 13 are prime factors of n, we can deduce that n must be divisible by all the products of the primes.
So we know that:
(3 × 5) = 15 must be a factor
(3 × 13) = 39 must be a factor
(5 × 13) = 65 must be a factor
(3 × 5 × 13) = 195 must be a factor
Use the handy prime factorization chart to visualize if you must.
Prime Factorization Chart |
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n |
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factor? |
factor? |
factor? |
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3 |
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13 |
Anything above 3, 5, 13 would be a factor of n. Any number in the above row would be a product of any combination of 3, 5, and 13. |
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The Greatest Common Factor (GCF)
This is the largest factor (divisor) for two numbers. For example, for 36 and 81 this would be 9 as the GCF. The primary use of GCF is to reduce fractions. For example, you would reduce 36/81 to 4/9 by dividing by the GCF of both numbers (9).
How to get the GCF:
- List the prime factors of each number.
- Multiply the factors both numbers share. If there are no common prime factors, then the GCF is 1.
Prime Factorization Chart |
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36 |
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12 |
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3 |
2 |
2 |
3 |
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Prime Factorization Chart |
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81 |
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27 |
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3 |
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9 |
3 |
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3 |
3 |
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Of the prime factors they share, there are two common ones, 3 and 3. The product is 9. So 9 is the greatest common factor.
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Tons of Multiples
Many students confuse factors and multiples. Just remember that any number has an infinite number of multiples. 4, 6, 8, 10 and 12 are all multiples of 2. Factors and multiples are inter-related. A number is a multiple of any one of its factors. For example, 24 is a multiple of any one of its factors, i.e., 24 is a multiple of 8.
The least common multiple (LCM) of several numbers is the smallest integer that is a common multiple of the several numbers. We use this to add and subtract fractions. The least common multiple for two or more numbers is the smallest number in the list of common multiples for the numbers.
You determine the LCM by making a list of multiples and seeing what the smallest number is that shows up on all of the lists -- that number will be the LCM. For example, if you wanted to add 1/4 + 1/5, you need the LCM of 20.
Summary
Greatest Common Factor (GCF) vs. Least Common Multiple (LCM) |
Greatest Common Factor (GCF)
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Least Common Multiple (LCM) |
Definition |
The Greatest Common Factor (GCF) is the largest factor that divides two numbers.
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A common multiple is a number that is a multiple of two or more numbers. Common multiples of 4 and 5 are 0, 20, 40, 60, etc..
The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both of the numbers (excluding zero). |
What is it used for? |
GCF can be used to reduce fractions to the simplest possible form.
What can 18/24 be reduced to?
Use GCF to determine 6 and divide both by 6.
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LCM is usually used to add and subtract fractions.
1/5 + 1/4 = ?
Use LCM to determine 20.
4/20 + 5/20 = 9/20 |
How to get it |
1. List the prime factors of each number.
2. Multiply the factors both numbers share.
If there are no common prime factors, the GCF is 1.
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1. Find the prime factors of each number
2. Find the prime factors NOT shared for each number.
3. Multiply each number by the factors NOT shared by the OTHER number. |
Examples |
| Example 1
What is the GCF of 18 and 24?
18 = 2 × 3 × 3
24 = 3 × 2 × 2 × 2
18 and 24 share 3 × 2. The greatest common factor is therefore:
3 × 2 = 6.
Example 2
What is the GCF of 2940 and 3150?
2940 = 2 × 2 × 3 × 5 × 7 × 7
3150 = 2 × 3 × 3 × 5 × 5 × 7
2940 and 3150 share 2,3,5,7 so the GCF is: 2 × 3 × 5 × 7 = 210
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Example 1
What are the LCM of 40 and 48?
Here are the prime factors of 40 and 48:
| 40 = 2 × 2 × 2 × 5 |
= 5 × 48 = 240 |
| 48 = 2 × 2 × 2 × 2 × 3 |
= 2 × 3 × 40 = 240 |
We can multiply 48 × 5 = 240
(5 is the only prime factor of 40 not shared by 48)
Or, we can multiply 40 by 2 × 3 = 240
Either way, we'll get 240.
Note: because you get the same with both, this makes it easy to double check.
Example 2
What about the LCM of the numbers 16 and 24? Factor:
| 16 = 2 × 2 × 2 × 2 |
= 2 × 24 = 48 |
| 24 = 2 × 2 × 2 × 3 |
= 3 × 16 = 48 |
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Write the LCM (Least Common Multiple) of 6 and 12. In other words, what would you use to add 1/6 + 1/12?
| Solution
The factor 2 × 3 is used only once because it occurs in both numbers. Thus, the LCM is (2 × 3 × 2) = 12. Or, what is the smallest number that both 6 and 12 go into? 12. 12 is the lowest number divisible by both 6 and 12.
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Find the LCM (Least Common Multiple) of the three numbers 20, 30 and 50. What would you use as a denominator to add 1/20 + 1/30 + 1/50 = ?
| Solution
Write each number as a product of prime numbers:
20 = 2 × 2 × 5
30 = 2 × 3 × 5
50 = 2 × 5 × 5
The factor 2 × 5 occurs in all three numbers; thus it is used only once.
The LCM is LCM = (2 × 5) × 2 × 3 × 5 = 300
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What's the smallest number divisible by 2, 3, 4, 5 and 6?
Solution
Let's try doing this through logic as opposed to the methods discussed above.
2 × 3 × 4 × 5 × 6 = 720
If something is divisible by 4, then it must be divisible by 2, so we can eliminate 2 from our list of factors.
2 × 3 × 4 × 5 × 6 = 360
If something is divisible by 6, then it must be divisible by 3 as well; so get rid of 3 from our list of factors:
2 × 3 × 4 × 5 × 6 = 120
Since 4 and 6 share 2's, we can get rid of a 2.
(2 × 2) × 5 × (2 × 3) = 120
We could get rid of one of those 2's:
(2 × 2) × 5 × (2 × 3) = 60
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ADVANCED QUESTIONS
If x = 40y and y is prime, then what is the greatest common factor of x and 18y, in terms of y.
Solution
18 = 3 × 3 × 2
40 = 2 × 2 × 2 × 5
Greatest common factor is 2.
The answer is 2y.
If all else fails in a question like this, Backsolve using the answer choices or Plug In numbers to test that 2y is the correct answer.
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Note: This is an introductory section for beginners or intermediates with the exception of the final cross multiplication section.
A fraction is one number divided by another number. It represents an unsolved division, such as 3/5.
- The numerator is the top number that represents the number of parts that are selected.
- The bottom number is the denominator. The denominator represents the number of equal parts into which an entity has been divided.
For example, if a garden is divided into 5 equal plots, 3 of the plots is 3/5 of the garden.
- A fraction that has 0 as its denominator (e.g., 5/0) is infinitely large and undefined (not a real number or integer).
- If the numerator is zero, then the fraction equals zero (e.g. 0/5 = 0).
- If the fraction has a numerator equal to the denominator (e.g., 5/5), the fraction is equal to 1.
Mixed Numbers are numbers that are equal to an integer plus a fraction.
The number 4 2/3 is the integer 4, plus the fraction 2/3. Any mixed number can be written as a fraction, and any fraction greater than 1 can be written as a mixed number.
- To express 4 2/3 as a fraction:
1) multiply 4 × 3
2) add the numerator to this product, 12 + 2 = 14
3) divide by the denominator: 4 2/3 = 14/3
- To convert the fraction 17/5 into a mixed number:
1)
divide by the denominator (17 divided by 5 is 3 with 2 remaining)
2)
add the remainder over the denominator: 17/5 = 3 2/5
Convert 4 5/7 into a fraction.
| Solution
We multiply 4 × 7 and obtain 28. Add 5 to this and obtain 33. Put this over 7, and we find 4 5/7 = 33/7
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Convert 79/9 into a mixed number.
| Solution
Divide 79 by 9 and obtain 8 with 7 remaining. Now add to the 8 the fraction 7/9 (8 7/9).
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Equivalent Fractions
A fraction that has a common factor in both numerator and denominator is equal to the fraction with the common factor canceled. The fraction 6/10 is equivalent to the fraction 3/5 since the common factor 2 occurs in both numerator and denominator of 6/10.
In fact, the following fractions are all equivalent:
3/5 = 6/10 = 9/15 = 12/20
A fraction that has no common factors in the numerator and denominator is said to be expressed in lowest terms.
A fraction with a negative numerator or denominator is equivalent to a negative fraction, that is, -3/5 = 3/-5 = - (3/5). If both numerator and denominator are negative, the fraction is positive, that is, -3/-5 = 3/5.
Reducing Fractions to Lowest Possible Terms
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Always solve fractions in the lowest terms
The GMAT always puts fractions in the lowest possible form. This means that with any fraction question, try to put it in the lowest form before going to the answer choices. The reason for this is obvious; if they didn't use the lowest form it would create confusion about the right answer because there could be two answers on a multiple choice test that are the same value. (1/2 and 2/4). |
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Use the Greatest Common Factor (GCF) to bring fractions down to the lowest possible form. Divide the numerator and denominator by the GCF to get the lowest possible amount.
Reduce 275/525 to lowest possible terms.
275 = 25 × 11
525 = 25 × 21
Cancel out the 25's to get 11/21.
Express 26/16 as a mixed number in lowest terms.
| Solution
The mixed number is found by dividing by 16 which gives us 1 with 10 remaining. Hence
26/16 = 1 10/16 = 1 5/8
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Multiplying and Dividing Fractions
To multiply fractions:
- Cancel out any common factors that appear in both numerators and denominators.
- Then, multiply all numerators to form one numerator and all denominators to form one denominator. This final fraction may then be written as a mixed number, if desired.
To divide fractions, say (x/y)/(a/b):
- Invert the divisor (the fraction a/b)
- Then, multiply the two fractions, i.e., (x/y)/(a/b) = (x/y) × (b/a).
Example:
(5/6) × (7/2) = (7× 5)/(6 × 2) = 35/12
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Shortcut: Canceling Fractions
You can speed up multiplying fractions by canceling out any common factors in the numerator and the denominator.
Look at this example:
4/5 × 5/8 =
The fives cancel out and the 4 cancels out and you are left with 1/2.
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Adding and Subtracting Fractions
To subtract one fraction from another, we simply add a negative fraction to a second fraction. Consequently, the rules for adding and subtracting are the same. The first step is to write the fractions such that each fraction has the same denominator. Then add or subtract the numerators. Then simplify.
1/3 + 1/4 = 4/12 + 3/12 = 7/12
To write all fractions with the same denominator, a quick choice is to multiply all denominators together. However, this may give a rather large denominator. To avoid a large denominator, we could find the least common denominator (LCD); it is the least common multiple (LCM) of all the denominators.
NOTE: Splitting the numerator is OK
(8 +10)/3 can be split into 8/3 + 10/3
(3x + y)/2x = 3x/2x + y/2x
You cannot, however, split the denominator:
WRONG:
How to Solve Complex Fractions
How do you solve a fraction of fractions? Simply invert the divisor and then turn it into a multiplication question.
Complex fractions are fractions of fractions. Complex fractions are annoying if you try to take them head on. But you don't have to. Instead, transform them into normal fractions according to this quick step: multiply the top fraction by the reciprocal of the bottom fraction.
Cross Multiplication
Cross multiplication is a simplified method
of solving for two fractions that are set equal to each other. Given
the problem:
Solve for x
1) Multiply x diagonally down by 6 to get 6x
2) Multiply 9 diagonally up by 4 to get 36
3)
Then we solve for x and get 6x = 36 or x = 6.
Although this is one way to solve such an equation, we can solve it more easily by using cross multiplication. In cross multiplication we multiply the numbers diagonally across from each other.
In the previous problem, we would simply multiply (9 × 4) and set it equal to 6 times x. So we have,
(9 × 4) = 6x
36 = 6x
6 = x
If 17/x = 85/15, then what is x equal to?
Solution
x = 3
Using cross-multiplication, we multiply 17 by 15 and 85 by x.
When we do this, we get 255 = 85x. Divide each side by 85, and we get 3 = x.
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Shortcut: Fractions Comparison
If you want to determine which fraction is bigger
2/3 or 400/700?
Multiply the numerator of one by the denominator of the other
2 × 700 = 1400
3 × 400 = 1200
1400 is larger, which means 2/3 is greatest.
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Note: This is an introductory section for beginners or intermediates.
Decimals are really fractions where the denominator is 10.
- the fraction 3/100 is written in decimal form as 0.03
- the fraction 223/10,000 is 0.0223.
If you draw this out in a number line it looks like this:
0..........1..........2...........3..........4.......|..5
This | = 4 8/10 or 4.8
Basic Definitions
1 = units digit
10 = tens digit
100 = hundreds digit
1000 = thousands digit
In the number
34,567
3 is ten-thousands digit
4 is thousands digit
5 is the hundreds digit
6 is tens digit
7 is units digits
The GMAT commonly asks questions where it makes specifications of the units digit.
If x is the units digit and y is the tens digit and z is in the hundreds digit, then show the three digit number zyx.
To solve, since z is in the hundreds digit, y is the tens digit, use 100z + 10y + x = zyx
How to Round Decimals
To round decimals: eliminate the digit on the right, but if it is 5 or greater, increase the right digit by one.
What is 6.785 rounded to the nearest
tenths? = 6.8
hundredths? = 6.79
Write this expression as a decimal:
| Solution
When we divide by 100,000, we move the decimal point 5 places to the left. Then 31.2 = 0.000312. |
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A mixed decimal is the sum of an integer and a decimal fraction, much like the mixed fraction. The integer 5 added to the decimal fraction 35/100 is 5 + 35/100 = 5 + 0.35 = 5.35. The mixed decimal, or a decimal fraction, is usually simply called a decimal. (Note: the leading 0 in 0.35 is simply convention; it does not have mathematical significance: 0.35 = .35).
Write the fraction 649/100 as a decimal.
| Solution
The numerator 649 can be written as the decimal "649.0". When we divide by 100, we move the decimal point 2 places to the left so that
649/100 = 6.49
Note: It is not necessary to write any zeros after the 9; that is, 6.490 is equivalent to 6.49.
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Express 0.075 as a fraction in lowest terms.
| Solution
The decimal is expressed as a fraction .075 = 75/1000 The denominator and numerator are factored resulting in:
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Adding and Subtracting Decimals
To add or subtract decimals, we write the decimals in a column with the decimal points aligned vertically.
- Combine the decimals with a plus sign.
- Combine the decimals with a minus sign.
- Subtract the sum of the negative decimals from the sum of the positive decimals.
In performing these tasks, we add zeros to the right of the decimal point so that each number has an entry in each column. For example, if we subtract 3.021 from 5, we write 5 as 5.000 so that there is an entry in each of the 3 places to the right of the decimal in both numbers.
Add 5 + 2.783 + 3.04.
| Solution
Write the decimals in a column, with zeros added if none exist:
..5.000
2.783
+ 3.040
10.823 |
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Compute 6.98 + 3.217 + 3 - 3.637
| Solution
There are two possible ways to solve this problem:
(1) Add the positive decimals and the negative decimals:
+ 6.980
+ 3.217
+ 3.000
- 3.637
...9.56 |
(2) Subtract the sum of the negative decimals from the sum of the positive decimals:
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Multiplying and Dividing Decimals
Multiply two decimals just like you would multiply two integers. The number of decimal places in the product is then equal to the total of the decimal places in the two decimals.
Compute 0.05 × 12.
| Solution
Set up the multiplication as though the decimals were integers:
12 × .05 = .60
The answer must have a total of 2 decimal places: .60
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To divide two decimals, move the decimal point in the divisor (the number doing the dividing) to the right so that the divisor is an integer. Move the decimal point in the dividend to the right the same number of places. Now perform the division, placing the decimal point in the answer directly above the decimal point in the dividend.
Compute .6/12
| Solution
Change .6 to .60 and divide .60 by 12.
.60/12 =.05
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Shortcut: Multiplying/Dividing by 10
Rule #1: If you multiply a decimal by 10, you would move the decimal 1 place to the right; 100, move 2 to the right; 1000, move 3 to the right, and so on.
For instance, 100,000 × 0.0054 = 540.
You see that there are 5 zeros in 100,000; therefore, we moved the decimal 5 places to the right.
Rule # 2: When you divide by 10, you move the decimal one place to the left; 100, move 2 to the left; 1000, move 3 to the left.
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1a. Exponent Rules
Fundamentals:
Common Exponents
These exponents commonly appear on the GMAT and should be memorized to save time.
| Squares |
Cubes |
Higher Powers |
Other Powers |
| 22 = 4 |
23 = 8 |
20 = 1 |
34 = 81 |
| 32 = 9 |
33 = 27 |
21 = 2 |
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| 42 = 16 |
43 = 64 |
22 = 4 |
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| 52 = 25 |
53 = 125 |
23 = 8 |
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| 62 = 36 |
63 = 216 |
24 = 16 |
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| 72 = 49 |
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25 = 32 |
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| 82= 64 |
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26 = 64 |
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| 92 = 81 |
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27 = 128 |
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| 102 = 100 |
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28 = 256 |
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| 112 = 121 |
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29 = 512 |
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| 122 = 144 |
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210 = 1024 |
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| 132 = 169 |
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| 142 = 196 |
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| 152 = 225 |
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Part of the benefit here is that you can go beyond the chart.
If 210 = 1024, then 2-10 = 1/1024 or 212 = 210 × 22 = 1024 × 4
The 16 Must Know Exponent Rules
| 1. |
z2 = z × z |
To the 2 power is called "squared." |
| 2. |
z3 = z × z × z |
To the 3 power is called "cubed." |
| 3. |
z0 = 1
-670 = 1 |
Anything to the 0 power always equals 1 |
| 4. |
z1 = z
-671 = -67 |
Anything to the 1 power = itself |
| 5. |
(1/2)2 = 1/4
(1/2)3 = 1/8
(1/2)4 = 1/16 |
The higher exponent you take a positive fraction, the smaller the number becomes.
(1/2)2 = 1/4 > (1/2)3 = 1/8 > (1/2)4 = 1/16 |
| 6. |
x-n = 1/(xn)
3-2 = (1/3)2 = 1/9
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A number taken to a negative exponent is a fraction gets put in the denominator with 1 on the top. |
| 7. |
(-2)2 = 4
(-2)4 = 16
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A negative number taken to any even exponent value is always a positive value. Why? The negatives cancel themselves out.
(-2)2 = -2 × -2 = +4
(-2)4 = 16 = -2 × -2 × -2 × -2 = +16 |
| 8. |
(-)(22) = -4
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Note: If you aren't squaring the negative sign, it stays negative (this rarely happens | |
