Wednesday, January 20, 2016

elementary number theory - What is the largest power of 2 that divides 200!/100!.


What is the largest power of 2 that divides 200!/100!.


No use of calculator is allowed. I had proceeded in a brute force method which i know regret.. I would like to know your methods.


Answer




Find highest power of 2 in 200! and 100!, using Legendre's formula


In 200!, highest power of 2


=200/2+200/4+200/8+200/16+200/32+200/64+200/128


=100+50+25+12+6+3+1=197


In 100!, highest power of 2


=100/2+100/4+100/8+100/16+100/32+100/64


=50+25+12+6+3+1=97


Now, just subtract the two, and we get 100 as the answer.


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