Find the smallest number by which 35672 must be divided so as to get a perfect cube

Let’s find out the prime factors of the given number,

8788 = 2 × 2 × 13 × 13 × 13

So, in this pair of triplets, two 2 are extra. Therefore to get the perfect cube we have to divide the given number by 2×2, which is 4.

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iii Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube: 72

Solution

72 Prime factors of 72=2×2×2×3×3 Here 3 does not appear in 3’s group. Therefore, 72 must be multiplied by 3 to make it a perfect cube. 2722362182923 1

Answer

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Hint: We will use the Prime factorization method to solve this problem. Prime factorization is finding prime numbers which when multiplied, gave  us the original number.

Complete step-by-step answer:

Given number is 8788.

Now, we’ll do the prime factorization of 8788

\[8788 = 2 \times 2 \times 13 \times 13 \times 13 \]

As we can see that the prime factor 2 doesn’t occur 3 times, so the given number is not a perfect cube.

Hence, we will divide 8788 by 4$(2 \times 2)$ to get quotient as a perfect cube

\[\Rightarrow \dfrac{8788}{4} = \dfrac{2\, \times\, 2\, \times\, 13\, \times\, 13\, \times\, 13}{4} \] 

\[\Rightarrow 2197 = 13 \times 13 \times 13 \]

2197 is a perfect cube.

Therefore, the smallest number by which 8788 must be divided to get the quotient as a perfect cube is “4”.

Note: A perfect cube is a number which is obtained on multiplying the same number thrice.

A perfect square is a number which is obtained by multiplying the same number twice.

Prime factorization is the process in which numbers will be broken down into sets of prime numbers which multiply together to result in the original number.

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