If d is the greatest common divisor of x and y, where x and y are

Question

If d is the greatest common divisor of x and y, where x and y are positive integer, and x>y, d is greatest common divisor of which of the following numbers?

I. d and x
II. y and xy
III. y and x-y

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

gmatophobia · Accepted Answer

Let's assume that the greatest common factor of x and y is d. x = d * X_1 y = d * Y_1 X_1 and Y_1 do not have any common prime factor (i.e. they are co-primes) https://gmatclub.com/forum/download/file.php?id=81667 Answer Choice Elimination I. d and x GCF( d, d*X_1 ) will be equal to d Hence, this statement is true II. y and xy GCF( d*Y_1, (d*X_1*d * Y_1) ) will be equal to d * Y_1 Hence, this statement is not true III. y and x-y GCF( d*Y_1, (d*X_1-(d * Y_1)) ) = GCF( d*Y_1, (d(X_1-Y_1)) ) ⇒ will be equal to d Hence, this statement is true I and III are therefore true. Option E GMAT-Club-Forum-5ienoxmg.png

Maxsparrow · Answer

I. d & x
Obviously correct, since x is divisible by d

II. y & xy
Obviously incorrect, since the GCD for these two are y

III y & x-y
Not so obvious, lets break it down
we know y is divisible by d, so the question is what comes of x-y

we know x > y
we know x = d q
we know y = d r
x - y = d (q-r)
Hence x - y is divisible by d.

Question is, is d the GCD for case III, or x - y yield something that is greater than d as GCD with y
It would be unlikely since 
we know d is GCD of x & y

ANS E

poojaarora1818 · Answer

Can we solve this question by plugging nos. where x>y
Let's consider x=24, Y=12 So, the GCD of X & Y is 12

I     d and x  (12 and 24) - True
II    y and xy(12 and 12*24=288) - False
III   y and x-y(12 and 24-12=12) -  True

Hence, the correct answer is option E

Bunuel · Answer

Learn how to use the forum properly: https://gmatclub.com/forum/download/file.php?id=83290 GMAT-Club-Forum-9719p4b3.png

verrona · Answer

I is correct because d is the greatest divisor of d itself.
II is incorrect because y should be the greatest common divisor of y and xy
III is correct because we can turn x and y into d*a and d*b (with a>b because x>y) then x-y = d(a-b). x-y must be less than x, while d is the greatest common divisor of x and y, d must be the greatest common divisor of x-y and y.

egmat · Answer

This is a classic GCD properties question that tests whether you understand what "greatest common divisor" really means. Let's work through it using a concrete example - this will make everything crystal clear.

Let's Use Smart Numbers

Since we need  x > y  and  d = GCD(x,y) , let me choose:
 
  x = 18  
  y = 12   
  d = 6  (which is indeed the GCD of 18 and 12)

Now let's test each option:

Option I: d and x

What's  GCD(6, 18) ?

Well, 6 divides both 6 and 18. In fact, since 6 divides 18, the greatest common divisor of 6 and 18 is simply 6 itself.

So  GCD(d, x) = d  ✓

Option II: y and xy

What's  GCD(12, 12 	imes 18) ? That's  GCD(12, 216) .

Here's the key insight: since  216 = 12 	imes 18 , we know that 12 divides 216 evenly. In fact,  y  always divides  xy  because  xy = x 	imes y .

This means  GCD(y, xy) = y  itself, not  d .

So  GCD(12, 216) = 12 , which is  y , not  d = 6  ✗

Option III: y and x-y

What's  GCD(12, 18-12) ? That's  GCD(12, 6) .

Since 6 divides 12 evenly ( 12 = 2 	imes 6 ), the GCD of 12 and 6 is simply 6.

So  GCD(y, x-y) = 6 = d  ✓

Why Does Option III Work?

Notice that if  d  divides both  x  and  y , then  d  must also divide their difference  (x-y) . This is because we can write  x = d 	imes m  and  y = d 	imes n  for some integers  m  and  n , which means  x - y = d(m-n) .

But here's what makes this special: there's a fundamental GCD property that states  GCD(a, b) = GCD(b, a-b) . The greatest common divisor stays the same when you replace one number with their difference.

Answer: (E) I and III only

Want to Master GCD Problems?

I've shown you the smart numbers approach here, but there's much more to learn. The  complete solution on Neuron by e-GMAT  includes the systematic framework for identifying GCD properties quickly, common traps students fall into (like confusing "common divisor" with "greatest common divisor"), and alternative solution methods. You'll also learn the general algebraic approach that doesn't require smart numbers at all. Plus, you can practice with  detailed solutions for hundreds of other official questions on Neuron  to build pattern recognition across all GMAT quant topics.

Hope this helps you see the logic behind GCD problems!

svsivakrishna · Answer

Dear @ bunuel, I came across this question on my mock 4, however, the correct answer choice is given as III only which is y and x-y,. However, here the answer is given as I and III. Please help me. if needed, i can send the screenshot of the mock 4 of this question.

Bunuel · Answer

Is it possible you are mixing this question with another one here https://gmatclub.com/forum/if-d-is-the- ... 37304.html ?

svsivakrishna · Answer

Ah, My bad, I am really sorry. Yes, The question I attempted is the one which is from the link. Thank you very much

If d is the greatest common divisor of x and y, where x and y are

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If d is the greatest common divisor of x and y, where x and y are

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