Question

Find the smallest value of n such that 120n is a square number

Answer 1

Answer:

n=30

Step-by-step explanation:

120*30=3600

Square root of 3600 is 60

3600/120 = 30

So n=30

Answer 2

We have to find the smallest value of n such that when it is multiplied by 120, it gives a square number.

To obtain that number, we need to find the prime factors of 120. Below is the factorization:

$\Large \begin{array}{c|c} \underline{\sf {2}}&\underline{\sf {\; \; 120 \; \; \: }} \\ \underline{\sf {2}}&\underline{\sf {\; \; 60 \; \; \: }}\\ \underline{\sf {2}}&\underline{\sf {\; \; 30 \; \; \: }} \\ \underline{\sf {3}}&\underline{\sf {\; \; 15 \; \; \: }} \\ \underline{\sf {5}}&\underline{\sf {\; \; 5 \; \; \: }} \\ & {\sf \; 1 \; \; }\end{array}$

120 = 2 x 2 x 2 x 3 x 5

The prime factors of a square number always exist in pairs. Here 2, 3 and 5 do not occur in pair. In order to have them in a pair, we multiply the number by 2 x 3 x 5 = 30, the smallest value of n.

Now, 120 x 30  = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5

             3600   = (2 x 2 x 3 x 5)²

                         = 60²

Therefore, 30 is the smallest of n such that 120n is a square number.