The square and the root

Answer: 0

Finding the square and square root of 49 (least efficient method)

7$\times$7 = 49 so the square root of 49 is 7.

49$\times$49 = 2401

2401$\div$7=343 remainder 0

Finding the square root of 49 but not the square

7$\times$7 = 49 so the square root of 49 is 7.

So 49$\times$49 = 7$\times$7$\times$7$\times$7, which is clearly a multiple of 7.

So the remainder is 0.

The most efficient method - finding neither of the numbers

The square root of 49 multiplied by itself is 49, so 49 is a multiple of the square root of 49.

So multiples of 49 are also multiples of the square root of 49.

49 squared is a multiple of 49.

So 49 squared is also a multiple of the square root of 49.

So the remainder when the square of 49 is divided by the square root of 49 is 0.