## find the smallest number by which 2560 must be multiplied so that the product a perfect cube. Give the answer with pic plz.​

Question

find the smallest number by which 2560 must be multiplied so that the product a perfect cube.
Give the answer with pic plz.​

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3 months 2021-07-19T22:26:59+00:00 2 Answers 0 views 0

1. Interesting! If you factorize 2560 we get

2560= 8^3 *5.

If we multiply the number by 5^2=25, the result will be 8^3* 5^3, which is the perfect cube.

So 25 is the smallest positive integer.

But just wait, the multiplying factor can be fraction as well!

If we multiply 2560 by (1/2560), the result is 1.

1 itself is a perfect cube. So answer can be 1/2560.

But wait again! Why it can’t be negative number? It can be.

If we multiply 2560 by (-2560 * 2560), the result will be – 2560*2560*2560 which is a perfect cube. So answer can be -2560*2560.

But wait, the story is not over yet. Any number of the form (- 2560*2560 * n^3), where n is a natural number, can be the answer. If 2560 is multiplied by this number the result is perfect cube & cube root will be -2560* n.

So there is no definite one answer.

Hope you followed what I have written.

25

Step-by-step explanation:

Prime factorising 2560, we get,

2560=2

9

×5.

We know, a perfect cube has multiples of 3 as powers of prime factors.

Here, number of 2’s is 9 and number of 5’s is 1.

So we need to multiply another 5

2

in the factorization to make 2560 a perfect cube.

Hence, the smallest number by which 2560 must be multiplied to obtain a perfect cube is 5

2

=25.