Problem
NT-B2-M05-P014 The Product \(i(i+1)\)
#14
★★★★☆ Level 4 of 5
Find the largest \(k\) such that \(2^k\mid\prod_{i=1}^{100} i(i+1)\).
Notice that the product equals \(100!\cdot101!\).
\(\prod_{i=1}^{100} i(i+1)=(1\cdot2\cdots100)(2\cdot3\cdots101)=100!\cdot101!\). Thus the exponent of \(2\) is \(v_2(100!)+v_2(101!)=97+97=194\). Answer: \(194\).
Hidden conversion of a product into factorials.