Unsure how to approach the problem besides using the fact that the LCM(a,b) * GCD(a,b) = a*b. I see the implication that the GCD(a,b)^3 = a * b. Perhaps it means a and b are different powers of the same prime decomposition?
Say a = 3 and b = 3^2: then gcd(a,b) = 3 -> gcd(a,b)^3 = 3^3 = 27 and a*b = 3 * 3^2 = 3^3
Then gcd(a,b)^3 = a*b since 3^3 = 3^3
I understand this implication but having trouble applying it generally to the prime decompositions of a and b.
Can it be said that the prime decomposition of b equals the prime decomposition of a^2 assuming b > a?
Thank you.
Let $a$ and $b$ be positive integers such that $\operatorname{lcm}(a,b)=\gcd(a,b)^2$. Now, let $p$ be any prime number, and let $n,m\ge 0$ be non-negative integers, such that $p^n$ is the largest power of $p$ dividing $a$ and $p^m$ is the largest power of $p$ dividing $b$.
Now, the largest power of $p$ which divides $\operatorname{lcm}(a,b)$ is $p^{\max(n,m)}$. Can you see why?
The largest power of $p$ which divides $\operatorname{gcd}(a,b)^2$ is $p^{2\cdot\min(n,m)}$. Can you see why?
Now, because prime factorizations are unique, it follows from $\operatorname{lcm}(a,b)=\gcd(a,b)^2$ that $\max(n,m)=2\min(n,m)$. In other words, if we look at $n$ and $m$, one of the two is twice the other one.
So we have found a necessary condition on the prime factorizations of $a$ and $b$. It is also sufficient, but I leave this as a challenge to you.
We are now also in a position to generate a lot of examples. Pick a random prime, say $7$. Pick a pair of integers where one is twice the other, say $2$ and $4$. Raise your prime to these powers. We find $7^2=49$ and $7^4 = 2401$.
Repeat this process as often as you like. Let's take $2$ as our next prime, and $5$ and $10$ as our exponents. We find $2^5=32$ and $2^{10}=1024$.
Let's take $19$ as our next prime, with exponents $1$ and $2$. We find $19^1=19$ and $19^2=361$.
Okay, now we have a few pairs of prime powers. From each pair, pick one. Say we pick $49$, $1024$ and $19$. Multiply them together to get $a$. We find $a=49\cdot 1024\cdot 19 = 953344$. Now take the other prime power of each pair, so $2401$, $32$ and $361$, and multiply those together to get $b$. So $b=2401\cdot 32\cdot 361=27736352$.
Finally, let's test. We find $$\operatorname{lcm}(953344,27736352)=887563264=29792^2 = \gcd(953344,27736352)^2,$$ like we wanted!