If the sum of two distinct positive integers is 50, then what is the

Question

If the sum of two distinct positive integers is 50, then what is the maximum possible value of the greatest common factor of these 2 integers ?

(A) 2
(B) 5
(C) 10
(D) 25
(E) 50

M37-13

chetan2u · Accepted Answer

hi...

here the choices will help us..
we are looking for the largest so lets start from highest
1)  50 
a+b=50
GCF cannot be GREATER than the number...NO
2)  25 ..
Only possibility is when both numbers are 25 ......25+25=50
numbers have to be DISTINCT...NO
3)  10 
10(a+b)=50
so a+b= 5..
various possibility a=1,b=4 that is 10 and 40
then 2, and 3 will give 20 and 30...YES

ans C

Bunuel · Answer

Official Solution:

If the sum of two distinct positive integers is 50, then what is the maximum possible value of the greatest common factor of these 2 integers ?

A.  2 
B.  5 
C.  10 
D.  25 
E.  50

Test options by starting with largest number.
 
E. The GCF cannot be 50, because GCF cannot be greater than any one of the numbers, thus cannot equal to their sum.
 
D. The GCF of 25 would mean that both numbers are 25, which is not the case because we are told that the numbers are distinct.
 
C. The GCF of 10 would mean that  x+y=10a+10b=50 , which gives  a+b=5 . If  (a, b)  is (1, 4) or (2, 3) (or vise-versa), then the two numbers,  x  and  y , will be (10, 40) or (20, 30) (or vise-versa). So, GCF of 10 is possible and since its the largest of the possible values among the options, then it must be correct..

Answer: C

davezhcoet · Answer

Bunuel  is this method correct?
Sum of two numbers = 50
HCF will be highest when LCM is least possible.
For LCM to be least one number should be equal to other but thats not possible since numbers have to be distinct so one number should be some multiple of another.
lets say one integer is a, then other number can be 2a; a+2a=50 not possible
a+3a=50 not possible. 
a+4a=50, it implies a=10 and other number 40,
hence HCF=10. Option C is the correct answer. I hope my way is OK

Bunuel · Answer

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

Kinshook · Answer

E 50 is not possible since there are 2 distinct positive integers

D 25 is not possible since they are distinct

C 10 is possible since 10a + 10b = 50 ; a+b=5 ; (a,b) = {(1,4),(2,3),(4,1)}

IMO C

Nups1324 · Answer

Hi chetan2u , I did not understand this step: 10(a+b)=50. Why did you multiply (a+b) with 10.? Thank you

Posted from my mobile device

chetan2u · Answer

We are looking for the GCF. 
So if 10, which is given in the choices, is the GCF, then the numbers are 10a and 10b. Their sum is 50 => 10a+10b=50...10(a+b)=50....a+b=5
So possible value of (a,b) are (1,4) and (2,3).
Numbers can be 10&40 or 20&30.

Crytiocanalyst · Answer

Rather than brute force finding it's simpler to eleminate the options

(C) 10
20 , 30 satisfies therefore let us hang on to it

(D) 25
x= 25 and y=25 since the digits has to be distinct therefore out

(E) 50
The 2 integers will overshoot the sum of 50 therefore out

Therefore IMO C

rvgmat12 · Answer

GCD is maximum if both numbers are equal 25 and 25

But it is given that the numbers are distinct

So, 10 + 40 = 50 - GCD = 10

C

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If the sum of two distinct positive integers is 50, then what is the

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