Let us calculate by which positive smallest number should be divided with the numbers below so that the quotient will be a perfect cube.

i. 7000 ii. 2662

iii. 4394 iv. 6750

v. 675

Let us recall what a perfect cube is.

A perfect cube is a number that is the cube of an integer.

(i). For 7000.

We need to factorize 7000. We get,

7000 = 7 × 2 × 5 × 2 × 5 × 2 × 5

Group these factors into three similar integers.

7000 = (2 × 2 × 2) × (5 × 5 × 5) × 7

To make it into perfect cube, divide it by 7 on both sides.

⇒ 1000 = (2 × 2 × 2) × (5 × 5 × 5)

Since, we are able to group it in three similar integers. This means, 1000 is a perfect cube.

Thus, smallest positive number that should be divided with 7000 is 7.

(ii). For 2662.

We need to factorize 2662. We get,

2662 = 2 × 11 × 11 × 11

Group these factors into three similar integers.

2662 = (11 × 11 × 11) × 2

To make it into perfect cube, divide it by 2 on both sides.

⇒ 1331 = (11 × 11 × 11)

Since, we are able to group it in three similar integers. This means, 1331 is a perfect cube.

Thus, smallest positive number that should be divided with 2662 is 2.

(iii). For 4394.

We need to factorize 4394. We get,

4394 = 2 × 13 × 13 × 13

Group these factors into three similar integers.

4394 = (13 × 13 × 13) × 2

To make it into perfect cube, divide it by 2 on both sides.

⇒ 2197 = (13 × 13 × 13)

Since, we are able to group it in three similar integers. This means, 2197 is a perfect cube.

Thus, smallest positive number that should be divided with 4394 is 2.

(iv). For 6750.

We need to factorize 6750. We get,

6750 = 5 × 5 × 5 × 3 × 3 × 3 × 2

Group these factors into three similar integers.

6750 = (5 × 5 × 5) × (3 × 3 × 3) × 2

To make it into perfect cube, divide it by 2 on both sides.

⇒ 3375 = (5 × 5 × 5) × (3 × 3 × 3)

Since, we are able to group it in three similar integers. This means, 3375 is a perfect cube.

Thus, smallest positive number that should be divided with 6750 is 2.

(v). For 675.

We need to factorize 675. We get,

675 = 5 × 5 × 3 × 3 × 3

Group these factors into three similar integers.

675 = (3 × 3 × 3) × 5 × 5

To make it into perfect cube, divide it by (5 × 5) on both sides.

⇒ 27 = (3 × 3 × 3)

Since, we are able to group it in three similar integers. This means, 27 is a perfect cube.

Thus, smallest positive number that should be divided with 675 is (5 × 5), i.e., 25.