Question

k is a non-zero whole number. Given that 6 * 54 * k is a perfect cube, write down the smallest value of k.

Answer 1

Step-by-step explanation:

Given Find the smallest non-zero whole number which can be multiplied by 112 to give a square number and a cube number

We need to find the prime factors of 112 by using prime factorization

So the factors are 2 x 2 x 2 x 2 x 7

Now 7 remains without a pair and therefore 7 should be multiplied to 112 to make it a perfect square.

So 112 x 7 = 784 = 2 x 2 x 7 = 28

So √784 = 28

Similarly 28 should be multiplied to 784 to make it a perfect cube.

So 784 x 28 = 21,952

So cube root of 21,952 will be 28.

Answer 2

Answer:

➡️ 6×54×k=perfect cube

➡️

$6 \times 54 = {6}^{2} \times {3}^{2}$

$k \times {6}^{2} \times {3}^{2} is \: a \: perfect \: cube$

➡️ Least value of k=3×6=18

So it will be a perfect cube