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?
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 4 must be the GCD and 504 must be the LCM.
the factors of 504 are
2*2*2*3*3*7
and the factor of 36 are
2*2*3*3
since the GCD of 36 and n is 4, n must have 2 factors of 2.
In addition to those two 2 factors that n has in common
with 36, n must also have another 2 factor as well
as a 7 factor, since 504 does and 36 doesn't. n doesn't need any 3
factors because 36 has two 3 factors. So n has the two 2 factors
in common with 36 and an additional 2 factor. That's three factors of
2 and one 7 factor, and so n = 2*2*2*7=56
Edwin
