BTGmoderatorLU wrote:Source: e-GMAT
Two positive integers \(a\) and \(b\) are divisible by \(5\), which is their largest common factor. What is the value of \(a\) and \(b\)?
1) The lowest number that has both integers \(a\) and \(b\) as its factors is the product of one of the integers and the greatest common divisor of the two integers.
2) The smaller integer is divisible by \(4\) numbers and has the smallest odd prime number as its factor.
The OA is E
Given: GCD of (a, b) = 5
We have to determine the values of \(a\) and \(b\).
Let's take each statement one by one.
1) The lowest number that has both integers \(a\) and \(b\) as its factors is the product of one of the integers and the greatest common divisor of the two integers.
The phrase, "The lowest number that has both integers \(a\) and \(b\) as its factors," means that the number if LCM of a and b.
So, LCM = (a or b)*GCD = (a or b)*5;
We know that the product of the numbers is equal to the product of their LCM and GCD, thus,
LCM * GCD = a * b
=> (a or b)*5*5 = a * b
Either a = 25 or b = 25. No unique answer. Insufficient.
2) The smaller integer is divisible by \(4\) numbers and has the smallest odd prime number as its factor.
Every integer is divisible by 1. The smallest odd prime number is 3. We already know that GCD = 5; thus, the smaller integer is also divisible by 5.
Thus, the smaller integer would be 1*3*5 = 15. It's divisible by 1, 3, 5 and 15 (only 4 numbers).
However, we do not have any information about the larger integer; moreover, we do not know which between the two a and b is smaller and larger. Insufficient.
(1) and (2) together
From both the statements, we know that one of the numbers is 25 and the other is 15, but which between a and b is 15 and which is 25, we are not sure. No unique answer. Insufficient.
The correct answer:
E
Hope this helps!
-Jay
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