The sum of the digits of the perfect square nearest to 59678 is:

Concept

The perfect square nearest to a number N is found by taking the square root of N, rounding to the nearest integer k, and comparing the two candidate squares k2 and (k±1)2. The one with the smaller absolute distance from N is the nearest perfect square.

Application

1. Estimate the square root: √59678 ≈ 244.3, so the candidate integers are 244 and 245.

2. Compute 2442 = 59536, with distance |59678 − 59536| = 142.

3. Compute 2452 = 60025, with distance |60025 − 59678| = 347.

4. Since 142 < 347, the nearest perfect square is 59536.

5. Add its digits: 5 + 9 + 5 + 3 + 6 = 28.

Cross-check

244 and 245 are consecutive integers whose squares bracket 59678, so no other integer square can be closer. The smaller distance (142) confirms the lower square is nearest, giving a digit sum of 28.