For nonzero integers \(a,b,c\), show that \[ \gcd(a,\operatorname{lcm}(b,c)) =\operatorname{lcm}(\gcd(a,b),\gcd(a,c)). \]
Solution
Given a prime \(p ,\) let \(\alpha , \beta ,\) and \(\gamma\) be the exponents of \(p\) in the prime factorization of \(a , b ,\) and \({ \mathit { c } } ,\) respectively. Then it will suffice to show that
\[ \operatorname* { m i n } \{ \alpha , \operatorname* { m a x } \{ \beta , \gamma \} \} = \operatorname* { m a x } \{ \operatorname* { m i n } \{ \alpha , \beta \} , \operatorname* { m i n } \{ \alpha , \gamma \} \} \ . \]
Without loss of generality, we may assume that \(\beta \leq \gamma ;\) in that case max \(\{ \beta , \gamma \} = \gamma\) and min \(\{ \alpha , \beta \} \le \operatorname* { m i n } \{ \alpha , \gamma \}\) Therefore the above equation is true because both sides are equal to min \(\{ \alpha , \gamma \}\)