Find the missing digit

Of course, one approach is to multiply out the number and see that we get $2 \times 653 \times 733 \times 977 = 935280146$ and hence the missing number is 7. That was how I created this problem, but it is not how I expect you to solve it.

Let's consider modulo 9. Let the number be expressed as $N$ . We have

$\begin{array} {l l l } N & \equiv 2 \times 653 \times 733 \times 977 & \pmod {9} \\ & \equiv 2 \times 5 \times 4 \times 5 \pmod{9} \\ & \equiv 2 \pmod{9} \end{array}$

The sum of all 10 digits is $0 + 1 + 2 + \ldots + 9 = 45$ . If the digit $a$ was not used, then the sum of the digits is $45 - a$ .

Since the remainder when divided by 9 is equal to the remainder of the sum of the digits when divided by 9, this tells us that $45 - a \equiv 2 \pmod{9}$ . Hence, we can conclude that $a = 7$ .