Problem
NT-B2-M05-P019 Factorial and a Square
#19
★★★★★ Level 5 of 5
Find the largest \(k\) such that \(36^k\mid150!\).
Factor \(36=2^2\cdot3^2\).
\(v_2(150!)=75+37+18+9+4+2+1=146\). \(v_3(150!)=50+16+5+1=72\). For \(36^k=2^{2k}3^{2k}\), we need \(2k\le146\) and \(2k\le72\), so \(k\le73\) and \(k\le36\). Answer: \(36\).
Composite base with equal prime exponents.