# How to find the missing digit?

A student calculated the value of $1 \times 2\times 3\times \cdots \times 2015\times 2016=2016!$

Then he took the summation of all digits of that answer !

He got $24135$ , but later he realized that he has missed a digit in the original answer.

What is the missing digit ?

Can anyone help ?

• $2016! \equiv x \pmod{9}$, what is $x$? – Daniel Fischer Jan 24 '16 at 16:56
• Is missing digit $3$ or $0$ – Archis Welankar Jan 24 '16 at 16:57
• By missed, do you mean it is deleted or that one of the given digits is incorrect? – Ross Millikan Jan 24 '16 at 16:58
• @RossMillikan deleted – Angelo Mark Jan 24 '16 at 17:12

See sum of digits of factorial are divisible by $9$ after $6!$ so for a number to be divisible by $9$ so the sum of digits should be divisible by $9$ so it should be $15+x=18$ thus $x=3$
• $6!=720$,$7!=5040$... ie divisible by $9$ so the nearest multiple of $9$ around $15$ is $18$ so it to be a multiple of $9$ we should add $3$ so missing digit is $3$ – Archis Welankar Jan 24 '16 at 17:18