Inspired by Saya Suka

Saya taught me that the rule of divisibility of 9 could give 2 possible answers for the digit, as in Chew-Seong's solution. Instead, if we use the rule of divisibility of 11, we are guaranteed a unique answer. So, let's apply this rule:

$1-3+a-7+6-7+4-3+6-8+0-0+0$ is a multiple of 11, so $-11 + a$ is a multiple of 11.
Since $0 \leq a \leq 9$ , hence $-11 \leq -11 + a \leq 2$ . The only multiple of 11 in this range is $-11$ .
So $-11 = -11 + a$ , or that $a = 0$ .

The number given $N$ is $15!$ , which is divisible by 9. Therefore, the sum of its digits must also be divisible by 9. The sum of digits without $a$ is 45, which is divisible by 9. From the answer options, $a$ is either 0 or 9.

Dividing $N(0)$ and $N(9)$ by 11, which is easy, we find that $N(0)$ is divisible by 11, while $N(9)$ is indivisible. Therefore, $a=\boxed {0}$ .

solution to this question is This Question.

Gunakan saja karakteristik dari 9 dan 11