Two digits were erased!

Number Theory Level 2

Calculated $15!$ , I wrote the result on the board.

$\huge 1\ \square\ 0\ 7\ 6\ 7\ 4\ 3\ 6\ \square \ 0\ 0\ 0$

Two of the digits, the second and the tenth, were deleted by my sister. What are these two digits?

Notation: $!$ denotes the factorial notation . For example, $8! = 1\times 2\times 3 \times \cdots \times 8$ .

9 and 2 4 and 0 4 and 8 3 and 8 7 and 4

1 solution

Riccardo Baldini
Jun 9, 2019

Called $a, b$ the two missing digits: $1a0767436b000$ , we can apply some divisibility rules to that number. For example, divisibility by 11 requires that the difference between $a+7+7+3+b+0=17+a+b$ and $1+0+6+4+6+0+0=17$ must be a multiple of 11: that is $a+b=11$ (since $a+b=0$ and $a+b=22$ are impossible).

From the available solution, we can discard $\{4,0\}$ and $\{4,8\}$ .

We can note that 15! contains 2 to the 11th power and 5 to the 3rd power, in particular the number must be divisible by $5^3*2^3 * 2^3=8000$ , so we need the digits 36b to be multiple of 8: the only way is for b=8.

So the answer is the couple a=3, b=8

Two digits were erased!

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