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14 May 2020, 08:31
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
56% (02:12) correct
44%
(02:27)
wrong
based on 106
sessions
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If x and y are positive integers, what is the remainder when y is divided by x?
(1) \(\frac{(x + y)^y}{(x + y)^x} = 149\)
(2) \(y^2 - x^2 = 149\)
Are You Up For the Challenge: 700 Level Questions
(1) \(\frac{(x + y)^y}{(x + y)^x} = 149\)
(2) \(y^2 - x^2 = 149\)
Are You Up For the Challenge: 700 Level Questions
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
14 May 2020, 08:54
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Bunuel
A very important fact here is 149 is a prime number (for GMAT we only need to know primes up to 100 however).
Nevertheless, let us approach this problem by utilizing this fact.
Statement 1:
We can write this as \((x+y)^{y - x} = 149\). Since 149 is not a power of any number, the only way to break it down into integers is \(149^1 = 149\). Then we can conclude the exponent is \(y - x = 1\) and the base must be \(x + y = 149\). Given these two equations we can solve for x and y, sufficient.
Statement 2:
Again we can write this as \((y - x)(y + x) = 149\), and since we can only break down 149 into 149 * 1, we must have \(x + y = 149\) and \(y - x = 1\). Sufficient.
Ans: D
14 May 2020, 09:21
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Bunuel
Given: x and y are positive integers
To find: remainder when y is divided by x.
For reference, I always write down this equation when I encounter remainder questions:
y = x*q + rem
Here, q is the quotient and rem is the remainder. We have to determine the remainder.
Statement 1: \(\frac{(x + y)^y}{(x + y)^x} = 149\)
Let's try to prime factorize 149. For this, we first find out the nearest square root, which is 12 here. We don't consider 13 because square of 13 is 169, a value which is greater than 149.
Therefore, we get that \((x + y)^(y-x) = 149\) {y-x is the power of x+y}
Clearly, 149 is not a square or cube or 4th power, etc. of any positive number ( x + y is a positive number ), thus, we can conclude that -
x + y = 149 and
y - x = 1
Since, y-x = 1, they are consecutive integers.
Therefore, the remainder when x is divided by y will always be 1 (since, y > x).
Sufficient
Statement 2: \(y^2 - x^2 = 149\)
From Statement 1, we already know that 149 is a prime number.
( y - x )( y + x ) = 149
x and y are positive integers, therefore, y + x = 149 and y - x = 1
This statement is also sufficient.
Correct answer - (D)
