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

Question

If x and y are positive integers, what is the greatest common factor of 11x and 22y?

(1) x is an odd integer.
(2) y = x + 22

WizakoBaskar · Answer

The HCF of 11x and 22y is 11*(HCF(x, 2y))
The question is basically about finding a unique value for HCF(x, 2y)

Statement 1 : x is an odd integer.
There are infinite possibilities. 
 Approach: Counter Example 
Example : x = 5 and y = 10. HCF(x, 2y) is HCF(5, 20) = 5
Counter example: x = 3 and y = 15. HCF(x, 2y) is HCF (3, 30) =3

No conclusive answer. Statement 1 alone is NOT sufficient.

Statement 2 : y = x + 22
HCF (x, 2y) = HCF (x, 2x + 44)
 Approach: Counter Example 
Example: x = 11; HCF (x, 2x + 44) is HCF (11, 22 + 44) = 11
Counter example: x = 5; HCF (x, 2x + 44) is HCF (5, 10 + 44) = 1

No conclusive answer. Statement 2 alone is NOT sufficient.

Combining the two statements : x is odd and y = x + 22
The example and counter example used in statement 2 satisfy both these conditions.

Despite combining the information in the two statements, we will have multiple answers for the HCF (x, 2y) and therefore, for HCF(11x, 22y)
Choice E is the answer

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

Archit3110 · Answer

from 1:
x can be 1,3,5... no relation of y given in sufficient

#2: y=x+22

value of x not given in sufficient

from 1& 2
x=1
y=23

so 11*1 and 22*23 = 11*2*23 ; GCF is 11
at x=3, y=25
 
 HCF (x, 2x + 44) is HCF (11, 22 + 44) = 11
 x = 5; HCF (x, 2x + 44) is HCF (5, 10 + 44) = 1

IMOE

Ypsychotic · Answer

Statement 1) x is an odd integer. Doesn't tell anything about y. y could be odd or even or a multiple of x, in which GCF changes. Insufficient.

Statement 2) y=x+22. For x=11, 11x and 22y will have GCF 121, for x =3, 11x and 22y will have GCF=11, for x= even, gcf=22. Different values. Insufficient.

(1)+(2)
As for x=3 and x=11, there are two different values as stated in statement 2. Insufficient.

Option E is correct.

GMATin · Answer

Hi  Bunuel  and other experts,

Is there an algebraic or logical approach to solve this question?

I have gone through the solutions other users have posted by picking numbers to come to a solution. I tried doing the same when combining Statements 1 and 2. However, picked x=3 and x=5 to check this. Of course after going through the answers I realised how x=11 and multiples of 11 proves that the two statements together are insufficient to come up with a unique answer, however I can't come up with general lessons from this question.

Can you please help me with the same.

Thanks!

Shobhit7 · Answer

HCF of 11x and 22y = definitely some positive power of 11, rest depends upon values of x and y.

S1: x is odd.
So, HCF of x and y will not have any power of 2 i.e HCF of x and y is odd. 
Beyond that, this statement is inconclusive as HCF value depends on common odd prime factors between x and y.
Insufficient.

S2. y=x+22. 
Case 1: x is odd: x and y can be co-primes (x=1, y=23) or have 11 as HCF (x=11,y=33)
Case 2: x is even: HCF of x and y be 2 (x=2, y=24) or 22 (y=22, y=44)
Insufficient.

S1+S2. 
x is odd. As stated in S2 / Case 2, HCF of x and y can be 1 or 11.
So, HCF of 11x and 22y can be 11 or 11^2.
Insufficient.

Ans E.

exc4libur · Answer

(1) x is an odd integer :  insufic.

x,y=1,3…gcf(11*1,22*3)=(11,11)=11
x,y=3,3…gcf(11*3,22*3)=(11*3,11*3)=11*3

(2) y = x + 22 :  insufic.

x=1…gcf(11x,22(x+22))=(11,22(23))=(11,11)=11
x=2…gcf(11x,22(x+22))=(11*2,22(24))=(11*2,11*2)=11*2

(1&2) :  insufic.

x=1…gcf(11x,22(x+22))=(11,22(23))=(11,11)=11
x=11…gcf(11x,22(x+22))=(11*11,22(33))=(11*11,11*11)=11*11

Answer (E)

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

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