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 Post subject: GMAT Number Theory (Data Sufficiency)
PostPosted: Sat Jun 19, 2010 1:02 am 
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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.


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 Post subject: Re: math (test 1, question 24): numbers theory, algebra
PostPosted: Sat Jun 19, 2010 1:04 am 
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Posts: 498
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


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 Post subject: Re: math (test 1, question 24): numbers theory, algebra
PostPosted: Fri Dec 10, 2010 7:08 am 
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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.


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 Post subject: Re: math (test 1, question 24): numbers theory, algebra
PostPosted: Wed Feb 29, 2012 4:59 am 
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A good tricky quetion!


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 Post subject: Re: math (test 1, question 24): numbers theory, algebra
PostPosted: Sun May 06, 2012 4:59 pm 
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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?


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 Post subject: Re: math (test 1, question 24): numbers theory, algebra
PostPosted: Mon May 07, 2012 3:19 am 
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 Post subject: Re: GMAT Number Theory (Data Sufficiency)
PostPosted: Thu Nov 29, 2012 7:28 pm 
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Joined: Tue Nov 27, 2012 7:29 am
Posts: 12
Why do you need the prime factors and is there a smart way to get to all the prime factors quickly?


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 Post subject: Re: GMAT Number Theory (Data Sufficiency)
PostPosted: Thu Nov 29, 2012 8:08 pm 
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Joined: Sun May 30, 2010 2:23 am
Posts: 498


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