Find the least number to be added to 5439 × 5447 to get a perfect square number.

Concept

For numbers close to a central value, use the difference of squares identity: (a − x)(a + x) = a2 − x2. The product of two numbers equidistant from a center a differs from the perfect square a2 by exactly x2. So, to convert such a product into a perfect square, add x2 to it.

Application

1. The two numbers are 5439 and 5447. Their average (center) is (5439 + 5447) / 2 = 5443.

2. Each number is 4 away from the center: 5439 = 5443 − 4 and 5447 = 5443 + 4.

3. So 5439 × 5447 = (5443 − 4)(5443 + 4) = 54432 − 42 = 54432 − 16, using a = 5443 and x = 4.

4. Since 54432 − 16 is 16 less than the perfect square 54432, adding 16 makes the expression exactly 54432.

Cross-check

Direct multiplication confirms this: 5439 × 5447 = 29,626,233, and 54432 = 29,626,249. The difference is 29,626,249 − 29,626,233 = 16, exactly matching the value derived above.

Result

The least number that must be added is 16.