The square and the root
What is the remainder when the square of 49 is divided by the square root of 49?
This problem is taken from the UKMT Mathematical Challenges.
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.