Find the sum of the digits of the smallest number which, when added to 695862,…

Concept

A perfect square is an integer of the form k2. To find the smallest number that must be added to a given N to reach a perfect square, locate the first perfect square that is greater than or equal to N: take m = ⌈√N⌉ (the smallest integer whose square is ≥ N); the required addition is then m2 − N. Summing the digits of that difference is a simple final step.

Application

1. Estimate the square root of N = 695862: √695862 ≈ 834.18, so the integers to test are 834 and 835.

2. Check 834: 8342 = 695556, which is less than 695862 — too small, so 8342 is not yet a perfect square at or above N.

3. Take the next integer m = 835: 8352 = 697225, the first perfect square at or above 695862.

4. Smallest number to add = 697225 − 695862 = 1363.

5. Sum the digits of 1363: 1 + 3 + 6 + 3 = 13.

Cross-check

Verify minimality: 8342 = 695556 < 695862 < 697225 = 8352, so 697225 really is the next perfect square and no smaller addition reaches a perfect square. Substituting back, 695862 + 1363 = 697225 = 8352, confirming the gap is 1363 and its digit sum is 13.