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).
10
A non-zero number is divisible by 10 if it ends in 0.
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)
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
Don't forget!
When a question asks... "how many factorsdoes... 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.
Example
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.
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
36
2
12
3
3
2
2
3
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 you can tell if any number is a factor of another number,
Is 36 a factor of 81?
Prime Factorization Chart
36
12
3
2
2
3
36 breaks into prime factors 2, 2, 3, 3
Prime Factorization Chart
81
27
3
9
3
3
3
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).
Example
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
n
factor?
factor?
factor?
3
5
13
Anything above 3, 5 and 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.
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
36
12
3
2
2
3
Prime Factorization Chart
81
27
3
9
3
3
3
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.
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)
Least Common Multiple (LCM)
Definition
The Greatest Common Factor (GCF) is the largest factor that divides two numbers.
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.
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.
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 of 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.
2940 and 3150 share 2,3,5,7 so the GCF is: 2 × 3 × 5 × 7 = 210
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
Example
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.
Example
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
Example
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
ADVANCED QUESTIONS
Example
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.
