What is the smallest number by which 1372 must be multiplied so that the product become a perfect cube?

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Algebra

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what is the smallest number by which 13500 should be multiplied so that the product is a perfect cube

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Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100

(i) We have 243 = 3 x 3 x 3 x 3 x 3

The prime factor 3 is not a group of three.∴ 243 is not a perfect cube.Now, [243] x 3 = [3 x 3 x 3 x 3 x 3] x 3

or, 729, = 3 x 3 x 3 x 3 x 3 x 3
Now, 729 becomes a [perfect cube

Thus, the smallest required number to multipkly 243 to make it a perfect cube is 3.(ii) We have 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Grouping the prime factors of 256 in triples, we are left over with 2 x 2.∴ 256 is not a perfect cube.Now, [256] x 2 = [2 x 2 x 2 x 2 x 2 x 2 x 2 x 2] x 2or, 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2i.e. 512 is a perfect cube.thus, the required smallest number is 2.(iii) we have 72 = 2 x 2 x 2 x 3 x 3

Grouping the prime factors of 72 in triples, we are left over with 3 x 3∴ 72 is not a perfect cube.Now, [72] x 3 = [2 x 2 x 2 x 3 x 3] x 3or, 216 = 2 x 2 x 2 x 3 x 3 x 3i.e. 216 is a perfect cube

∴ The smallest number required to multiply 72 to make it a perfect cube is 3.

(iv) We have 675 = 3 x 3 x 3 x 5 x 5

Grouping the prime factors of 675 to triples, we are left over with 5 x 5

∴ 675 is not a perfect cube.Now, [675] x 5 = [3 x 3 x 3 x 5 x 5] x 5Now, 3375 is a perfect cubeThus, the smallest required number to multiply 675 such that the new number is a perfect cube is 5.(v) We have 100 = 2 x 2 x 5 x 5The prime factor are not in the groups of triples.

∴ 100 is not a perfect cube.Now, [100] x 2 x 5 = [2 x 2 x 5 x 5] x 2 x 5or, [100] x 10 = 2 x 2 x 2 x 5 x 5 x 51000 = 2 x 2 x 2 x 5 x 5 x 5Now, 1000 is a perfect cubeThus, the required smallest number is 10