Question

Suppose r is a positive integer such that the lcm(11, r) = (11r)^2. What is the smallest possible value for r?

Answer 1

let \(\gcd(12,r)=d\) and \(\text{lcm}(12,r)=m\) you hav the equation\[m^3=(12r)^2\]and we know that \(md=12r\) so\[m=\frac{12r}{d}\]put this in the original equation\[\left(\frac{12r}{d} \right)^3=(12r)^2 \]\[r=\frac{d^3}{12}\]\(d=6\) will provide the smallest \(r=18\) (why?)

Answer 2

easier approaches??