Problem
NT-B1-M01-P019 Square Divisibility
#19
★★★☆☆ Level 3 of 5
Let \(a,b\) be positive integers. Prove that if \(a^2\mid b^2\), then \(a\mid b\).
Compare the exponent of each prime in \(a\) and in \(b\).
Let \(p^\alpha\) be the power of a prime \(p\) in \(a\), and \(p^\beta\) its power in \(b\). From \(a^2\mid b^2\), we get \(2\alpha\le2\beta\), hence \(\alpha\le\beta\). This holds for every prime \(p\), so all prime powers of \(a\) occur in \(b\). Therefore \(a\mid b\).
The problem reinforces the coordinate view of prime exponents.