If the 8-digit number 16131p13 is divisible by 9, then the minimum value of p…

Concept

A whole number is divisible by 9 if and only if the sum of its digits is a multiple of 9. This follows from the fact that every power of ten leaves remainder 1 on division by 9, so a number and its digit sum leave the same remainder.

So, to fix a missing digit, add the known digits first and then choose the digit that lifts the total to a multiple of 9.

Application

1. The digit sum of 16131p13 is 1 + 6 + 1 + 3 + 1 + p + 1 + 3.

2. The seven known digits add to 1 + 6 + 1 + 3 + 1 + 1 + 3 = 16, so the digit sum equals 16 + p.

3. Divisibility by 9 requires 16 + p to be a multiple of 9. Since p is a single digit, 16 + p lies between 16 and 25, and 18 is the only multiple of 9 in that range.

4. Solving 16 + p = 18 gives p = 2. No smaller digit works, and reaching the next multiple of 9 would need 16 + p = 27, that is p = 11, which is not a digit.

Cross-check

For p = 2 the number is 16131213, whose digit sum is 18, and the division is exact: 16131213 = 9 × 1792357.

The other offered digits give digit sums that are not multiples of 9:

• p = 3 gives digit sum 19.

• p = 4 gives digit sum 20.

• p = 7 gives digit sum 23.

Therefore the minimum value of p is 2.