The least number which must be added to 3931 to make it a perfect square, is

Concept

To turn a number into a perfect square by adding the smallest possible amount, locate the two consecutive integers whose squares straddle the number. If the number is not itself a perfect square, the smallest perfect square that is greater than or equal to it is the square of the larger integer, and the least number to add is (that next perfect square) minus the given number.

Application

1. Estimate the square root of 3931. Since 622 = 3844 and 632 = 3969, the value of 3931 lies between 622 and 632.

2. Because 3931 is not itself a perfect square, the smallest perfect square greater than or equal to 3931 is 632 = 3969.

3. The least number to add = 3969 − 3931 = 38.

Cross-check

Check: 3931 + 38 = 3969, and 632 = 3969, so the sum is indeed a perfect square.