If the 8-digit number 40569p14 is divisible by 3, then the maximum value of p is

Concept

A whole number is divisible by 3 exactly when the sum of its digits is divisible by 3.

For an unknown digit, add the known digits first, then choose a digit from 0 to 9 that makes the total digit sum a multiple of 3.

Application

1. The known digits of 40569p14 sum to 4 + 0 + 5 + 6 + 9 + 1 + 4 = 29.

2. Therefore, the full digit sum is 29 + p. Divisibility by 3 requires 29 + p to be a multiple of 3.

3. Since 29 leaves remainder 2 on division by 3, p must leave remainder 1. The possible digits are 1, 4, and 7.

4. The maximum of these admissible digits is 7.

Cross-check

With p = 7, the digit sum is 29 + 7 = 36, which is divisible by 3. The only larger digits are 8 and 9; they give digit sums 37 and 38, neither divisible by 3.

Hence, the maximum value of p is 7.