If 5472 = 9, 6342 = 6, and 7584 = 6 under the same rule, what is the value of…

Concept

The digital root of a positive integer is obtained by repeatedly adding its digits until one digit remains.

Adding the digits once preserves the eventual digital root, so the reduction can continue from the intermediate sum.

Application

1. For 5472, the digit sum is 5 + 4 + 7 + 2 = 18, and 1 + 8 = 9.

2. For 6342, the digit sum is 6 + 3 + 4 + 2 = 15, and 1 + 5 = 6.

3. For 7584, the digit sum is 7 + 5 + 8 + 4 = 24, and 2 + 4 = 6.

4. For 9236, the digit sum is 9 + 2 + 3 + 6 = 20, and 2 + 0 = 2.

Cross-check

A positive integer and its digital root have the same remainder modulo 9. Since 20 leaves remainder 2 when divided by 9, the required value is 2.