The number N is the produce of the first 1000 positive integers and can be written as 1000!. We say "1000 factorial."This is, N=1000!=1000x999x998x997x ... x3x2x1. N is divisible by 5, 25, 125, 625, ... Each of these factors is a power of 5. That is 5=5^1, 25=5^2, 125=5^3,625=5^4, and so on. Determine the largest power of 5 that divides N.

Question

Mathematics

Zackary Powell

19 September, 13:45

The number N is the produce of the first 1000 positive integers and can be written as 1000!. We say "1000 factorial."This is, N=1000!=1000x999x998x997x ... x3x2x1. N is divisible by 5, 25, 125, 625, ... Each of these factors is a power of 5. That is 5=5^1, 25=5^2, 125=5^3,625=5^4, and so on. Determine the largest power of 5 that divides N.

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Dominic Mejia · Answer 1

From 1000 to 1, there are 200 numbers that are divisible by 5, which means that 1000! is divisible by at least 5^200. Then there are 40 numbers that are divisible by 25 (5^2), so this adds another 5^40. Then there are 8 numbers divisible by 125 (5^3), so this adds another 5^8. Finally, 1 number is divisible by 625 (5^4), so this adds 5^1.

Therefore, the total sum is 5^ (200+40+8+1) = 5^249. Therefore, the largest N = 249.