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If x and y are positive integers, what is the greatest commo

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by crackverbal » Wed Oct 26, 2016 10:28 pm
Hi alanforde800maximus,

The question here tests us on the concept of co-primes. Let me first explain the concept of co-primes before moving on to solve the question.

Two integers a and b are said to be co-prime if the only positive integer that divides both of them is 1. That is, the only common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor(factor) being 1.

for e.g. 4 and 5 are co-prime since the only common factor they share is 1. Similarly 4 and 9 are also co-prime to each other. Two even numbers can never be co-prime as they will always share 2 as a factor.

Two consecutive numbers on the number line will always be co-prime. This is a very important result and is exactly what the question here tests us on.

The question here is a value DS where x and y are given to be positive integers.

Statement 1 : When x is divided by y, the remainder is 1 :

Now there are only two possibilities here

1. The numerator x is always 1 and the denominator y can be any positive integer greater than 1 (1/4, 1/5, 1/103.....). Since x = 1 and y is any positive integer greater than 1, the GCD of x and y will always be 1.

2. The numerator x is always one greater than the denominator y, in other words x and y are consecutive positive integers on the number line where x > y (5/4, 7/6, 143/142....). Since x and y are consecutive positive integers they will always have a GCD of 1. Statement 1 is sufficient

Statement 2 : x^2 - 2xy + y^2 = 1 :

Simplifying we get (x - y)^2 = 1 -----> x and y are consecutive positive integers on the number line (x = 3 and y = 2 or x = 5 and y = 6). Here again the GCD of x and y will always be 1. Statement 2 is sufficient.

OA : D

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by [email protected] » Thu Oct 27, 2016 11:16 am
Hi alanforde800Maximus,

This DS question can be solved by a combination of TESTing VALUES and doing a bit of Algebra. We're told that X and Y are POSITIVE INTEGERS. We're asked for the Greatest Common Factor (GCF) of X and Y.

1) When X is divided by Y, the remainder is 1.

This Fact tells us that X = Y +1. This piece of information involves a relatively rare math concept - and one that you probably will not see on the GMAT: when dealing with two consecutive, positive integers, the GCF will always be 1 (meaning that NO other integers will divide evenly into both numbers). Even if you don't know that rule though, it's easy enough to prove. Here are some examples:

X=4, Y=3... X is divisible by 1, 2 and 4... Y is divisible by 1 and 3... the GCF is 1
X=5, Y=4... X is divisible by 1 and 5... Y is divisible by 1, 2 and 4... the GCF is 1
X=6, Y=5... X is divisible by 1, 2, 3 and 6... Y is divisible by 1 and 5... the GCF is 1
Etc.
Thus, the answer to the question is ALWAYS 1.
Fact 1 is SUFFICIENT

2) X^2 - 2XY + Y^2 = 1

This quadratic equation can be 'reverse-FOILed' into...

(X - Y)(X - Y) = 1
(X - Y)^2 = 1
(X - Y) = 1 or -1
Since X and Y differ by 1 or -1, and they're both positive integers, we have the same situation that we have in Fact 1 (above).
Fact 2 is SUFFICIENT

Final Answer: D

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