is an integer Formula used: We need to find the largest n such that 100n divides 100 ! 100=22×52⇒100n=22n×52n So, we must find maximum n such that both 22n and 52n divide 100 ! Calculations: Powers of 2 in 100! ⇒[100∕2]+[100∕4]+[100∕8]+[100∕16]+[100∕32]+[100∕64] ⇒50+25+12+6+3+1=97 Powers of 5 in 100! ⇒[100∕5]+[100∕25]+[100∕125] ⇒20+4+0=24 Now: 22n must ≤297⇒2n≤97⇒n≤48.5 And: 52n must ≤524⇒2n≤24⇒n≤12 So, the limiting factor is powers of 5⇒n≤12 The largest value of n is 12 .
