Problem
NT-B2-M05-P012 Power of \(30\)
#12
★★★★☆ Level 4 of 5
Find the largest \(k\) such that \(30^k\mid200!\).
Factor \(30=2\cdot3\cdot5\).
\(v_2(200!)=197\), \(v_3(200!)=66+22+7+2=97\), \(v_5(200!)=40+8+1=49\). Since \(30^k\) requires \(k\) copies of each prime \(2,3,5\), the answer is \(\min(197,97,49)=49\).
Shows that the rarest prime is limiting.