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

Answers ( )

    0
    2021-07-19T22:28:01+00:00

    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.

    0
    2021-07-19T22:28:57+00:00

    Answer:

    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.

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