Find the value of Z for which the number 417Z8 is divisible by 9.

Concept

A whole number is divisible by 9 if and only if the sum of its digits is a multiple of 9.

For a missing digit, add the known digits first, then choose the digit that makes the total reach a multiple of 9.

Application

1. Let S be the sum of the digits of 417Z8.

2. The known digits give 4 + 1 + 7 + 8 = 20, so S = 20 + Z.

3. Since Z is a digit, S lies from 20 to 29. The only multiple of 9 in this interval is 27, so 20 + Z = 27.

4. Subtracting 20 from both sides gives Z = 7.

Cross-check

For Z = 7, the number is 41778 and its digit sum is 4 + 1 + 7 + 7 + 8 = 27, which is divisible by 9.

Therefore, Z = 7.