Question

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

Answer 1

Hint: In order to this question, to find smallest number by which each of the following numbers must be divided to obtain a perfect cube, we will first write the prime factors of each of the given numbers separately, and then try to find out the number which is not groups in their triplets, and then divide the given number by their ungrouped number, and that is the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

Complete step-by-step answer:
We will first find the prime number of each of the given numbers and then divide the number that makes the number in a perfect cube. And that’s the smallest number by which the given numbers must be divided to obtain a perfect cube separately.
(i) 81
Prime factors of 81 \[ = 3 \times 3 \times 3\times 3\]
Here one factor 3 is not grouped in triplets.
Therefore 81 must be divided by 3 to make it a perfect cube.

(ii) 128
Prime factors of 128 \[ = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
Here one factor 2 does not appear in a 3’s group.
Therefore, 128 must be divided by 2 to make it a perfect cube.

(iii) 135
Prime factors of 135 \[ = 3 \times 3 \times 3 \times 5\]
Here one factor 5 does not appear in a triplet.
Therefore, 135 must be divided by 5 to make it a perfect cube.

(iv) 192
Prime factors of 192 \[ = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\]
Here one factor 3 does not appear in a triplet.
Therefore, 192 must be divided by 3 to make it a perfect cube.

(v) 704
Prime factors of 704 \[ = {\text{2}} \times 2 \times 2 \times 2 \times 2 \times 2 \times 11\]
Here one factor 11 does not appear in a triplet.
Therefore, 704 must be divided by 11 to make it a perfect cube.



Note: The factor of a given number that is a prime number is called a prime factor. Factors are numbers that are multiplied together to produce a new number. The factor of a given number that is a prime number is called a prime factor. To put it another way, prime factoring is the process of determining which prime numbers multiply to produce the original number.