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

Hint: To answer this problem first find all divisors of 36 then add these all divisors in 500. Once we write the term by adding all divisors to 500 then try to find which one of them is divisible by 36. Then apply the property the multiplication of LCM and HCF is equal to the product of the two numbers.

Complete step-by-step answer:
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 \[GCD = {\text{ }}\left( {504 - 500} \right) = 4\]
Now apply the property of LCM and GCD
\[
\Rightarrow {GCD \times LCM = 36 \times n} \\
\Rightarrow {4 \times 504 = 36 \times n} \\
  {n = 56}
\]

Note: Here in this solution GCD means greatest common divisor which is exactly the same as the HCF highest common factor so students aren't confused.