If d is the greatest common divisor of x and y, where 1

Question

If d is the greatest common divisor of x and y, where 1 < d < x and 1 < d < y, then d is greatest common divisor of which of the following?

I. x and 1 
 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, II and III

Bunuel · Accepted Answer

Let's evaluate the options:

I. x and 1

The greatest common divisor of x and 1 is naturally 1. If d were the GCD of x and 1, that would imply that d = 1. However, we are told that 1 < d. So, option I is incorrect.

II. y and xy

The GCD of y and xy would be y. But we know that 1 < d < y, so d cannot be the GCD here. Option II is incorrect.

III. y and x-y

If a number is a divisor of two positive integers, it would also be a divisor of their difference and sum. Thus, d, being a divisor of both x and y, would also divide their difference, x - y. Therefore, d must be the GCD of y and x - y.

Alternatively, expressing x and y as dm and dn, where m and n are co-prime, we get x - y = d(m - n). The GCD of dn and d(m - n) would be d itself.

Answer: C.

RishS94 · Answer

X= K*d & Y= J*d.. so X-Y =d*(J-K) Divisible by d

Kinshook · Answer

@cjcalero24

If d is the greatest common divisor of x and y, where 1 < d < x and 1 < d < y, then d is greatest common divisor of which of the following?

I. x and 1; Greatest common divisor (x,1) = 1; Since d>1; Incorrect 
 II. y and xy; Greatest common divisor (y, xy) = y*Greatest common divisor(1,x) = y*1 = y; Since d<y; Incorrect 
 III. y and x-y; Greatest common divisor (y,x-y) = Greatest common divisor (y,x) = d

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

IMO C

soma7 · Answer

simple example 5 and 15:
1- GCD(5,1) -> 1 INCORRECT
2- GCD(15,75) -> 15 INCORRECT
3- GCD(15,10) -> 5 CORRECT

SatvikVedala · Answer

x = da
y = db
GCD(x, 1) => GCD(da,1) = 1
GCD(da, da.db) => GCD(da, d2.a.b) = da
GCD (db, d(a-b)) => GCD(db, d(a-b)) = d

CS0509 · Answer

But this pair of numbers will not work because d has to be less than x. for these two, the d is equal to x (5), right?

KarishmaB · Answer

When GCD of two numbers is given, it is helpful to think of the numbers as say x = 10a and y = 10b assuming the GCD d = 10 (for convenience) We are given that the numbers x and y are both greater than d so they are greater than 10. So a and b are co-prime positive integers greater than 1. I. x and 1 GCD of 10a and 1 is 1 only. Hence it is not d. II. y and xy The GCD of y and xy must be y only. The GCD of a number and its multiple is the number itself. Since d is not equal to y, GCD of y and xy is also not d. Now the answer is clear. It has to be III only. Answer (C) Here is a post on GCD and LCM: https://anaprep.com/number-properties-s ... roperties/ Let's still evaluate III III. y and x-y y = 10b x-y = 10a - 10b = 10(a-b) The GCD of 10b and 10(a-b) must be 10 because a and b are co-prime. So (a-b) will have no common factor with b. For example, (5-2) = 3 cannot have any factor common with 2 since 5 is co-prime with 2.

AbhishekP220108 · Answer

This is how I approached

Take X= dk1, Y=dk2 as it is given that d is factor of X & Y.

so in X & 1, 1 is HCF. In Y & XY, Y is HCF & in Y & X-Y, d is hcf, because X-Y=dK1-DK2 which is X-Y=d(k1-k2). Hence only III

sujoykrdatta · Answer

Since d is the GCD of x and y, x and y must be multiples of d
=> x = d*p and y = d*q; where p and q have no common factors

Checking the options: 
I. Clearly, the GCD is 1, hence incorrect
II. Clearly, the GCD is y, hence incorrect 
III. x - y = d(p-q) => GCD of dq and d(p-q) is d, hence correct 
(Note: Since p and q have no common factors, q and p-q also have no common factors. If they did, p and q would also have that factor common, which would contradict with the fact that d is the GCD of p and q)

Answer C

Aboyhasnoname · Answer

Bunuel  Just a quick question here..if d is the GCD of x and y...then it d would just be a divisor of y and x+y or the greatest common divisor... ?

Bunuel · Answer

If d = gcd(x, y), then:
 
 d divides x and y by definition. 
 Since d divides both, d also divides x + y. 
 Now, suppose g = gcd(y, x + y). Then g must divide y and x + y. But that means g also divides (x + y) - y = x. So g divides x and y. 
 Therefore, g ≤ d. But we also know d divides y and x + y, so d ≤ g. 
 Together: g = d.

Examples:
 
 x = 12, y = 18, then gcd(12, 18) = 6. Then gcd(18, 30) = 6. 
 x = 8, y = 12, then gcd(8, 12) = 4. Then gcd(12, 20) = 4. 
 x = 21, y = 15, then gcd(21, 15) = 3. Then gcd(15, 36) = 3.

In every case, gcd(y, x + y) = gcd(x, y).

bumpbot · Answer

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If d is the greatest common divisor of x and y, where 1 < d < x and 1

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