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GMAT Number Theory (Data Sufficiency)
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Author:  questioner [ Sat Jun 19, 2010 1:02 am ]
Post subject:  GMAT Number Theory (Data Sufficiency)

What is the value of the two-digit number TU, where T represents the tens digit and U represents the units digit?
(1) T × U² = 196
(2) 3T + 5U = 47

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient
to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the
question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to
answer the question, meaning that further information would be needed to answer the question.


(A) Since T and U are digits in a number, they must both be integers between 0 and 9 inclusive. In fact, T must be between 1 and 9 inclusive, since it is the tens digit.

Statement (1) tells us that (T) × (U²) = 196. From here, we should take the prime factorization of 196 to help us solve for T and U:
196 factors into 2 × 2 × 7 × 7.

Since we need to collect the prime factors in such a way that we get a digit times the square of another digit, U² must be 7²
and T must be 4. If U² were 2², then T would be 49, which is impossible if T is to be a digit. So T = 4, U = 7, and the number TU = 47. Statement (1) is sufficient.

Using Statement (2), we need to substitute possible values for
T and determine which one(s) yield possible values for U (remember, U must be a positive integer less than 10). If T = 4, then U = 7, and if T = 9, then U = 4. Therefore, TU could be 47 or 94. Statement (2) is not sufficient.


Since Statement (1) is sufficient and Statement (2) is not, the
correct answer is choice (A).
-------------

It's not mentioned if TU is +ve integer or a -ve integer. Now consider U = -7, then U² becomes +ve and TU²2 is still 196. So for statement 1 we can have two solns -47 or +47.

Author:  Gennadiy [ Sat Jun 19, 2010 1:04 am ]
Post subject:  Re: math (test 1, question 24): numbers theory, algebra

U is a digit and it can not be negative. The only values it can take are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Author:  Gennadiy [ Fri Dec 10, 2010 7:08 am ]
Post subject:  Re: math (test 1, question 24): numbers theory, algebra

As many questions, this one has various approaches. One more I'd like to share with you.

When we consider the statement (1) by itself, we can solve it as the equation, where U is unknown:

T × U² = 196
U² = 196 / T
U = 14/(√T)

U is a digit. Therefore √T must be a divisor of 14: 1, 2, 7 or 14. So T must be 1, 4, 49 or 196.

T can NOT be 49 or 196, since it is a digit.
If T was 1 then U would be 14, which would be impossible, since U had to be a digit.

Therefore the only possible option for T is 4. U = 7. The integer is 47.

As you can see the main fact we use is still that T and U are digits, as in the question explanation. But we look at the statement (1) from different angle.

Author:  questioner [ Wed Feb 29, 2012 4:59 am ]
Post subject:  Re: math (test 1, question 24): numbers theory, algebra

A good tricky quetion!

Author:  paulozac [ Sun May 06, 2012 4:59 pm ]
Post subject:  Re: math (test 1, question 24): numbers theory, algebra

Are you saying that, once we validate that statement I is sufficient, if the values don't match statement II, we can consider II insufficient?

Author:  Gennadiy [ Mon May 07, 2012 3:19 am ]
Post subject:  Re: math (test 1, question 24): numbers theory, algebra

paulozac wrote:
Are you saying that, once we validate that statement I is sufficient, if the values don't match statement II, we can consider II insufficient?
NO. We found the definite value using the statement (1) and it will satisfy the statement (2) as well. But if we use the statement (2) by itself there will also be other possible values of TU. Thus we will NOT be able to know the definite value of TU based on the statement (2) alone.

In other words, we can conclude that the statement (2) is insufficient if we show that two different outcomes, based on the statement (2), are possible. By two different outcomes we understand those that answer the main question differently.

In case of this problem we have:
Quote:
If T = 4, then U = 7, and if T = 9, then U = 4. Therefore, TU could be 47 or 94. Statement (2) is not sufficient.
As you can see the two possible numbers, 47 and 94, both satisfy statement (2) but are different values of the number TU. So the question cannot be answered definitely using the statement (2) alone.

Author:  Robert.Delane [ Thu Nov 29, 2012 7:28 pm ]
Post subject:  Re: GMAT Number Theory (Data Sufficiency)

Why do you need the prime factors and is there a smart way to get to all the prime factors quickly?

Author:  Gennadiy [ Thu Nov 29, 2012 8:08 pm ]
Post subject:  Re: GMAT Number Theory (Data Sufficiency)

Robert.Delane wrote:
Why do you need the prime factors and is there a smart way to get to all the prime factors quickly?
We need the prime factorization of 196 to see what the digits T and U can be. Especially U, because we have T × U² = 196 and thus U² must be a product of pairs, for example (2 × 2), or (7 × 7), or (2 × 2) × (7 × 7). In this case the latter does not fit, because U is a digit.

How to get prime factors of N quickly?
1. Start with dividing by 2, 5, or (2 × 5) = 10.
These are easy to spot. If the number is even, then divide it by 2 until it becomes odd.
If the number ends in 5 or 0, then it is divisible by 5.
N = (2 × … × 2)(5 × … × 5) × M

2. Then check if 3 is a factor.
If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.
N = (2 × … × 2)(5 × … × 5) × (3 × … × 3) × K

3. Now you have the remaining number, K. All you are left to do is to check other primes from 7 and up till √K
Why do we check below √K only? Because if there is a prime divider greater than √K, let's say x, then there will be a divider K/x (below √K).

But the third step won't be hard in gmat. The number K will be either a square of a prime number, or divisible by 7 or 11, for example 77.

EXAMPLE:
Factorize number 548856000.
1) Divide by 10
548856000 = 548856 × 10³

2) 548856 is even, so divide by 2 until it becomes odd.
548856 = 274428 × 2
= 137214 × 2²
= 68607 × 2³

3) 68607 is divisible by 3, because 6 + 8 + 6 + 0 + 7 = 27 is divisible by 3.
68607 = 22869 × 3
= 7623 × 3²
= 2541 × 3³
= 847 × (3 × 3 × 3 × 3)

4) Try dividing by 7
847 / 7 = 121, so 847 = 121 × 7
and 121 is 11² so combine all of the above:

548856000 = 10³ × 2³ × (3 × 3 × 3 × 3) × 7 × 11²
don't forget to separate 10³

548856000 = (2 × 2 × 2 × 2 × 2 × 2) × (3 × 3 × 3 × 3) × 5³ × 7 × 11²

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