Estonian Mathematical Olympiad

As $\text{lcm}(m,n)=2024=8\cdot 253$ and $8=2^3$, at least one of the numbers $m$ and $n$ is divisible by $8$. Since $8\mid 96=m-n$, the other one must also be divisible by $8$. Both $m$ and $n$ are divisors of $2024$. All divisors of $2024$ that are divisible by $8$ are $8$, $88$, $184$ and $2024$. The only two of these with difference $96$ are $184$ and $88$. A straightforward check shows that $\text{lcm}(184,88)=\text{lcm}(8\cdot 23,8\cdot 11)=8\cdot 23\cdot 11=2024$ indeed.

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Alternative solution.

As $\text{lcm}(m,n)=2024$, both $m$ and $n$ are divisors of $2024$. All divisors of $2024$ are 1, 2, 4, 8, 11, 22, 23, 44, 46, 88, 92, 184, 253, 506, 1012, 2024. As $m-n=96$, we must have $m>96$. This observation cuts out all cases except $m=184,m=253,m=506,m=1012$ and $m=2024$. As differences between $253$, $506$, $1012$ and $2024$ are larger than $96$, it suffices to check the options $m=184$ and $m=253$ which imply $n=88$ and $n=157$, respectively. As $157\nmid 2024$, the only solution can be $(m,n)=(184,88)$. An easy check shows that $\text{lcm}(184,88)=\text{lcm}(8\cdot 23,8\cdot 11)=8\cdot 23\cdot 11=2024$ indeed.