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Found problems: 2

1980 Putnam, A2
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$ depending only on $r$ and $s.$
least common multiple elementary-number-theory casework number theory

ICMC 4, 5
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$. [i]Proposed by Harun Khan[/i]
elementary-number-theory powers of 2 number theory modular arithmetic