Find the smallest integer m 0 such that x 2573133m is a pe

Find the smallest integer m > 0 such that x = (2^5)(7^3)(13^3)(m) is a perfect square. Fully explain your answer

we know any number is called perfect square if it\'s square root is integer.

which means we need same number repeated Even number of times.

2^5 means 2*2*2*2*2 in which we can make two Even pairs of 2*2, 2*2 , (which leaves 2) ...(i)

7^3 means 7*7*7 in which we can make one Even pair of 7*7 , (which leaves 7) ...(ii)

13^3 means 13*13*13 in which we can make one Even pair of 13*13, (which leaves 13) ...(iii)

using equation (i), (ii), (iii) we find that , we are left with 2*7*13

to get perfect square , we must complete their pair, means we need 2*7*13 extra

which means m=2*7*13=182

Hence 182 is the required answer