If x and y are positive integers, and 1 is the greatest common divisor

Question

If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y?

A. 1

B. 2

C. 5

D. 6

E. Cannot be determined

sanjitscorps18 · Answer

The GCD of 2x and 3y would still be 1 as we are multiplying x and y with two more primes that have nothing in common other than 1. 4 Primes would again have nothing in common except 1.

Let say x = 5 and y = 7, then 2x = 2 x 5 = 10 and 3y = 3 x 7 = 21 would still have no common factors other than 1

IMHO Option A

Apologies, doing too many things at one time. Option A is not correct but Option E is. Here we are only given that these are co-primes and not primes individually

GMATWhizTeam · Answer

Solution:

This question has a trap.

Most of us might think the answer is A. However, it is not. Let's understand how.

The question tells us that " 1 is the greatest common divisor of x and y ". This means that  x and y are co-prime to each other . They are  not necessarily prime individually .

Taking a couple of examples:

let  x=1  and  y=2 . So GCD of  2x  and  3y  or GCD of  2  and  6  is  2 .

let  x=3  and  y=1 . So GCD of  2x  and  3y  or GCD of  6  and  3  is  3 .

So we see that we cannot determine the GCD of  2x  and  3y  just by knowing that  x  and  y  are co-primes.

Hence the right answer is    Option E   .

bv8562 · Answer

Concept :
 HCF  of  TWO CONSECUTIVE INTEGERS  is  1 . But the  reverse is NOT true  i.e if the HCF of two integers is 1 it is not necessary that they are consecutive.

Given, x and y are positive integers, and 1 is the greatest common divisor of x and y. Based on the concept above there are two possibilities:

1. Either, both are consecutive integers, or
2. Both are prime integers

1. Both are consecutive integers: x=2 and y=3 then 2x=4 and 3y=9, HCF(2,3)=1 and HCF(4,9)=1
2. Both are prime integers: x=3 and y=5 then 2x=6 and 3y=15, HCF(3,5)=1 and HCF(6,15)=3

Multiple answers, therefore our answer is  E .

sarthak1701 · Answer

If the GCF is not the same in 2 sets of numbers, it cannot be determined as x and y can take virtually any positive value.

Say x = 5 and y = 6, GCF is 1

2x = 10 and 3y = 18, here the GCF is 2

But if x = 9 and y = 10

Then 2x = 18 and 3y = 30, here the GCF is 6.

Therefore the GCF can vary across sets and it cannot be determined. Ans E.

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If x and y are positive integers, and 1 is the greatest common divisor

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If x and y are positive integers, and 1 is the greatest common divisor

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