Divisible by Every Common Divisor of \(a\) and \(b\)

Number Theory Level 4

Find the number of pairs of positive integers \((a,b)\) with \(1\leq a < b \leq 100\) such that

there is at least one positive integer \(m\) with \(a<m<b\) such that \(m\) is divisible by every common divisor of \(a\) and \(b.\)

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