Find all positive integers $a$ and $b$ satisfying $\gcd (a,b)=10$ and $\operatorname{lcm} (a,b)=100$ simultaneously.

Question

Find all positive integers $a$ and $b$ satisfying $$\gcd (a,b)=10$$ and $$\operatorname{lcm} (a,b)=100$$ simultaneously.

Answer 1

WLOG

$\dfrac aA=\dfrac bB=10;(A,B)=1$

$[a,b]=[10A,10B]=10[A,B]=100$

$\implies[A,B]=?$ with $(A,B)=1$

Answer 2

From the given,

$$a=10n,b=10m$$ where $n,m$ are relative primes and $$10nm=100.$$

Hence from the factorizations of $10$, the solutions

$$10,100;20,50;50,20;100,10.$$