Question

Find the smallest number by which 432 can be divided so that the quotient is a perfect cube.

Answer 1

The smallest number by which 432 can be divided so that the quotient is a perfect cube is 2.

Concept :

Perfect cube:

The cube of a number is that number raised to power 3.

As we know that 2³ = 8 , 3³ = 27, 4³ = 64 , 5³ = 125 , 6³ = 216….

The numbers  8, 27, 64, and 125 are called perfect cubes.

The natural number n is perfect if there exists a natural number m such that m × m × m = n.

Given :

Number is = 432

To find :

The smallest number by which 432 can be divided so that the quotient is a perfect cube.

Solution :

Step 1 :

Resolving 432 into prime factors:

432 =2 × 2 × 2 × 2 × 3 × 3 × 3

Step 2 :

Grouping the factor in triplets of equal factors

432 = (2 × 2 × 2) × 2 × (3 × 3 × 3)

Here, one 2 is left which does not form a triplet.

To make 432 a cube, we have to eliminate 2. so, we divide  432 by 2 to make it a perfect cube.

$\frac{432}{2} = (2 \times 2 \times 2) \times \frac{2}{2} \times (3 \times 3 \times 3)$

216 = (2 × 2 × 2) x (3 × 3 × 3) is a perfect cube of 6.

Hence, the smallest number by which 432 should be divided to make it a perfect cube is 2.

Learn more on Brainly :

Which of the following is both a perfect square and a perfect cube? 16 64 125 144

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Question 3 Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704

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