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

i. 675 ii. 200

iii. 108 iv. 121

v. 1225

Let us recall what a perfect cube is.

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

(i). For 675.

We need to factorize 675. We get,

675 = 3 × 3 × 3 × 5 × 5

Group these factors into three similar integers. We have

675 = (3 × 3 × 3) × 5 × 5

To make it into a perfect cube, multiply both sides by 5.

We have

675 × 5 = (3 × 3 × 3) × 5 × 5 × 5

Now, group these factors into three similar integers.

3375 = (3 × 3 × 3) × (5 × 5 × 5)

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

Thus, smallest positive number that should be multiplied with 675 is 5.

(ii). For 200.

We need to factorize 200. We get,

200 = 2 × 2 × 2 × 5 × 5

Group these factors into three similar integers. We have

200 = (2 × 2 × 2) × 5 × 5

To make it into perfect cube, multiply both sides by 5.

We have

200 × 5 = (2 × 2 × 2) × 5 × 5 × 5

Now, group these factors into three similar integers.

1000 = (2 × 2 × 2) × (5 × 5 × 5)

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

Thus, smallest positive number that should be multiplied with 200 is 5.

(iii). For 108.

We need to factorize 108. We get,

108 = 2 × 2 × 3 × 3 × 3

Group these factors into three similar integers. We have

108 = (3 × 3 × 3) × 2 × 2

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

We have

108 × 2 = (3 × 3 × 3) × 2 × 2 × 2

Now, group these factors into three similar integers.

216 = (3 × 3 × 3) × (2 × 2 × 2)

We are able to group it in three similar integers. This means, 216 is a perfect cube.

Thus, smallest positive number should be multiplied with 108 is 2.

(iv). For 121.

We need to factorize 121. We get,

121 = 11 × 11

We can’t group it into three similar integers. So, multiply both sides by 11.

121 × 11 = 11 × 11 × 11

Now, group these factors into three similar integers.

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 multiplied with 121 is 11.

(v). For 1225.

We need to factorize 1225. We get,

1225 = 5 × 5 × 7 × 7

We can’t group it into three similar integers. So, multiply both sides by (5 × 7).

1225 × 5 × 7 = 5 × 5 × 7 × 7 × (5 × 7)

Now, group these factors into three similar integers.

42875 = (5 × 5 × 5) × (7 × 7 × 7)

Since, we are able to group it in three similar integers. This means, 42875 is a perfect cube.

Thus, smallest positive number that should be multiplied with 42875 is (5 × 7), i.e., 35.