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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 ?

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

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  • $\begingroup$ can you explain more please ? $\endgroup$ – Angelo Mark Jan 24 '16 at 17:11
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    $\begingroup$ $6!=720$,$7!=5040$... 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 $3$ $\endgroup$ – Archis Welankar Jan 24 '16 at 17:18
  • $\begingroup$ Oh thank you very much ! $\endgroup$ – Angelo Mark Jan 24 '16 at 17:19
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    $\begingroup$ Your welcome!!.. $\endgroup$ – Archis Welankar Jan 24 '16 at 17:22

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