Find the smallest number which when multiplied with 53240 will make the product a perfect cube

Last updated at Sept. 11, 2018 by

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Example 3 Is 53240 a perfect cube? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube? We see that 53240 = 2 × 2 × 2 × 5 × 11 × 11 × 11 Here, 5 does not occur in triplets ∴ 53240 is not a perfect cube. So, we divide by 5 to make triplet So, our number becomes 53240 × 𝟏/𝟓 = 2 × 2 × 2 × 5 × 11 × 11 × 11 × 𝟏/𝟓 = 2 × 2 × 2 × 11 × 11 × 11 Now, it becomes a perfect cube. So, we divide 53240 by 5 to make it a perfect cube

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Last updated at Aug. 22, 2018 by Teachoo

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Text Solution

Solution : Prime factors of `137592 = 2^3 xx 3^3 xx 7^3 xx 13` Here it is clear that when we multiply `137592` by the number `7 × 13^2` we get, `2^3 xx 3^3xx 7^2 xx 13 xx 7 xx 13^2 = (2 xx 3 xx 7 xx 13)^3 `⇒a perfect cube. So, the smallest number is `7 xx 13^2= 1183` and the cube root of product `= root(3){(2 xx 3 xx 7 xx 13)^3}` `= 2 xx 3 xx 7 xx 13 = 546` Therefore, the smallest number is `1183` and the cube root of the product is `546`.

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