Question

By what numbers should each of the following be divided to get a perfect square? Also, find the number whose square is the new number. 

 3174

Answer 1

3174 = 2 x 3 x 23 x 23

Grouping them into pairs of equal factors:
3174 = 2 x 3 x (23 x 23)
The factors, 2 and 3 are not paired. For a number to be a perfect square, each prime factor has to be paired. Hence, 3174 must be divided by 6 (2 x 3) for it to be a perfect square.
The new number would be (23 x 23). 
Hence, the number whose square is the new number is 23.