Question

What is the value of the positive integer $n$ for which the least common multiple of $36$ and $n$ is $500$ greater than the greatest common divisor of $36$ and $n$ ?

• A. $42$
• B. $52$
• C. $56$
• D. $64$

Answer 1

The correct option is C $56$

The divisors of $36$ are $1,2,3,4,6,9,12,18,36$
$500$ more than these are $501,502,503,504,506,509,512,518,536$
The LCM of $n$ and $36$ must be among these.
All multiples of $36$ end with an even digit, so that narrows the LCM of $n$ and $36$ down to $502,504,506,512,518,536$
$504$ is the only one of those which is a multiple of $36$
So, $LCM=504$ and $GCD$= (504-500)=$4$
$GCD\times LCM=36\times n$

$4\times 504=36\times n$

$n=56$