$\DeclareMathOperator{\lcm}{lcm}$$\DeclareMathOperator{\mcd}{mcd}$Found all couples $a,b$ in $\mathbb Z$ which verify $\gcd(a,b)=10$ and $\lcm(a,b)=100$.
I've tried to decompose $a$ and $p$ in prime numbers and I've taken the property $\gcd(a,b)\cdot \lcm(a,b)=a\cdot b$ but when I've done that, I only found one solution which is $a=b=5^2\, 2^2$.
$$\gcd(a,b)=10\to a=10m,b=10n,\gcd(m,n)=1\\ lcm(a,b)=lcm(10m,10n)=10mn=100\to mn=10$$ since $\gcd(m,n)=1$ we have different cases:
Case 1:
$m=1,n=10 \quad \to\quad a=10,b=100$
Case 2:
$m=2,n=5 \quad \to\quad a=20,b=50$
Case 3:
$m=5,n=2 \quad \to\quad a=50,b=20$
Case 4:
$m=10,n=1 \quad \to\quad a=100,b=10$