Trailing Zeros in Factorial Sum

Number Theory Level pending

Find the number of trailing 0's in the following expression: 1 ! + 2 ! + 3 ! + 4 ! + ⋅ ⋅ ⋅ + 2016 ! + 2017 ! 1!+2!+3!+4!+\cdot \cdot \cdot + 2016! + 2017!

  • The symbol ! ! is the Factorial Notation.


The answer is 0.

Muhammad Rasel Parvej by Muhammad Rasel Parvej , 27, Bangladesh

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2 solutions

Marta Reece
Jul 18, 2017

5 ! 5! does have a zero at the end, as it contains 2 × 5 2\times5 , as do all factorials above that.

3 ! + 4 ! = 6 + 24 = 30 3!+4!=6+24=30 also end in a zero.

However 1 ! + 2 ! = 3 1!+2! =3 so it ends in 3 3 , as does the entire expressions, since only zeroes are being added to the 3 3 .

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Expression 1 ! + 2 ! + 3 ! + 4 ! 1! + 2! + 3! + 4! It has no zero if it does not have zero in a row from right. So the unit number is not a zero.

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