Question

Express 48 as the product of its prime factors. Hence or otherwise find the smallest possible integer value of x for which is 48x is a perfect square

Answer 1

3

Explanation:

prime factors of 48 :

48 ÷ 2 = 24, 0 remainder

24 ÷ 2 = 12, 0 remainder

12 ÷ 2 = 6, 0 remainder

6 ÷ 2 = 3, 0 remainder

3 ÷ 2 has remainder, so next prime number

3 ÷ 3 = 1, 0 remainder; reached 1, so process ends.

for a perfect square we need an even number of every prime factor, because the square root will divide the exponents of all factors by 2.

like 36, which is 2×2×3×3. and the square root is then 2×3 (6).

48 has 4 factors of 2 and 1 factor of 3.

we need to multiply it with one time 3. then we will have 4 factors of 2 and 2 factors of 3 :

48×3 = 144

and the square root of 144 is 12 (2×2×3 meaning 2 factors of 2 and one of 3).

Answer 2

48 =
`2^(4) . 3`
48x is a perfect square -> it is divisible by 48.

since the prime factor 3 doesn't have a pair, it must be multiplied to 48 to get the smallest square

-> 48x = 144
x = 3

and yes, 144 = 12 or -12 squared