IMO 1975 SL 6

Origin: USS

Problem

Let A be the sum of the digits of the number 1616 and B the sum of the digits of the number A. Find the sum of the digits of the number B without calculating 1616.

Solution

Let us denote by C the sum of digits of B. We know that 1616 \equivA \equiv B \equivC (mod 9). Since 1616 = 264 = 26\cdot10+4 \equiv24 \equiv7, we get C \equiv7 (mod 9). Moreover, 1616 < 10016 = 1032, hence A cannot exceed 9 \cdot 32 = 288; consequently, B cannot exceed 19 and C is at most 10. Therefore C = 7.