Is 53240 a perfect cube?

To check whether a given number is a perfect cube or not, we need to find its prime factors and check the power of each prime factor.

Prime factorization of 53240:

53240 = 2 × 2 × 2 × 5 × 7 × 11 × 13

Now, we need to check the power of each prime factor.

Power of 2 = 3 (since there are three 2's)

Power of 5 = 1

Power of 7 = 1

Power of 11 = 1

Power of 13 = 1

Since the power of 2 is not a multiple of 3, 53240 is not a perfect cube.

What is the smallest natural number by which it should be divided so quotient is perfect cube?

To make the given number a perfect cube, we need to divide it by the smallest possible number such that the power of each prime factor becomes a multiple of 3.

In this case, we need to divide 53240 by 2 (since the power of 2 is 3 and we need to make it a multiple of 3).

53240 ÷ 2 = 26620

Now, let's find the prime factorization of 26620:

26620 = 2 × 2 × 5 × 7 × 19 × 19

Power of 2 = 2

Power of 5 = 1

Power of 7 = 1

Power of 19 = 2

We need to divide 26620 by 19 (since the power of 19 is 2 and we need to make it a multiple of 3).

26620 ÷ 19 = 1400

Now, let's find the prime factorization of 1400:

1400 = 2 × 2 × 2 × 5 × 5 × 7

Power of 2 = 3

Power of 5 = 2

Power of 7 = 1

Therefore, the smallest natural number by which 53240 should be divided to get a perfect cube is 2 × 19 = 38.

When we divide 53240 by 38, we get 1400, which is a perfect cube.