Can you stop checking for factor pairs when you find a pair that repeats? Explain

Solution:

Consider numbers which are Squares, cubes, fourths, fifths of some natural numbers.

For example, Starting from squares of some natural number: 4,9,16,25,36,49,64 ...

and their factors, which are 4 = 1 * 2*1*2

9=1*3*1*3

16=2*1*2*1*2*1*2*1

25=1*5*1*5

36=2*3*2*3

64=2*4*2*4

Now coming to cubic numbers

8 = 1 * 2*1*2*1*2

27 = 1*3*1*3*1*3

3125 = 3 * 5*3*5*3*5

So, the numbers whose factor pairs repeats are square, cubic, fourth, Fifth, Sixth and ... higher powers of natural Numbers.

→→→So, we just need to check that whether those numbers whose factor pairs repeats are Squares, cubes or higher powers of natural numbers.