Question

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100

Answer 1

Answer:

3

2

3

25

10

Step-by-step explanation:

We need to find smallest number by which each of the following numbers must be multiplied to obtain a perfect cube

243 = 3 * 3 * 3 * 3 * 3

= 3³ * 3²

if We multiply by 3 Then it will be a perfect cube

243 * 3 =  3³ * 3² * 3 = 3³ * 3³ = 9³

256 = 2 * 2 * 2 * 2 * 2 * 2  * 2 * 2

= 2³ * 2³ * 2²

if We multiply by 2 Then it will be a perfect cube

256 * 2 = 2³ * 2³ * 2²* 2 = 2³ * 2³ * 2³  = 8³

72 = 2 * 2 * 2 * 3 * 3   = 2³ * 3²

if We multiply by 3 Then it will be a perfect cube

72 * 3  = 2³ * 3² * 3 = 2³ * 3³ = 6³

675 = 5 * 5 * 5 * 5  = 5³ * 5

if we multiply by 5² = 25 then  it will be a perfect cube

675 * 25 = 5³ * 5  * 5² = 5³ * 5³ = 25³

100 = 2 *  2 * 5 * 5  = 2² * 5²

if we multiply by 2 * 5 = 10 then  it will be a perfect cube

100 * 10 = 2² * 5²  * 2 * 5 =  2³ * 5³ = 10³

Answer 2

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(i) Prime factorisation of 243, is:

243 = 3 × 3 × 3 × 3 × 3 = 33 × 3 × 3

Here, number 3 is required to make 3 × 3 a group of three, i.e., 3 × 3 × 3

Thus, the required smallest number to be multiplied is 3.

ii) Prime factorisation of 256, is:

256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23 × 2 × 2

Here, a number 2 is needed to make 2 × 2 a group of three, i.e., 2 × 2 × 2 Thus, the required smallest number to be multiplied is 2.

(iii) Prime factorisation of 72, is:

72 = 2 × 2 × 2 × 3 × 3 = 23 × 3 × 3

Here, a number 3 is required to make 3 × 3 a group of three, i.e. 3 × 3 × 3

Thus, the required smallest number to be multiplied is 3.

(iv) Prime factorisation of 675, is:

675 = 3 × 3 × 3 × 5 × 5 = 33 × 5 × 5

Here, a number 5 is required to make 5 × 5 a group of three to make it a perfect cube, i.e. 5 × 5 × 5

Thus, the required smallest number is 5.

(v) Prime factorisation of 100, is:

100 = 2 × 2 × 5 × 5

Here, number 2 and 5 are needed to multiplied 2 × 2 × 5 × 5 to make it a perfect cube, i.e., 2 × 2 × 2 × 5 × 5 × 5

Thus, the required smallest number to be multiplied is 2 × 5 = 10.

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