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You can factor 540 into prime factors. Then, for each prime factor that doesn't appear 3, 6, 9, ... times, add additional factors to complete a multiple of 3. These factors will make up the number "k".

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Q: What is the smallest integer k such that 540k is a cube number?

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If by cube you mean perfect cube (a cube of an integer), then no, and the nearest perfect cube is 81.

its 81

The cube root of this number is one more than the smallest prime

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A perfect cube is the cube of an integer (whole number). This means that, for n to be a perfect cube, n = x3, x∈ℤ Eg. ±1 [=(±1)3], 8 [=(±2)3], ±27 [= (±3)3], etc.

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The integer is 26

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64 = 4 cubed and 8 squared.

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If by cube you mean perfect cube (a cube of an integer), then no, and the nearest perfect cube is 81.

its 81

A cube is any number multiplied by itself three times, eg 2 cubed = 2³ = 2×2×2 = 8; 1.5³ = 1.5×1.5×1.5 = 3.375 A perfect cube is an integer (whole number) that is the cube of an integer, eg 8 is a perfect cube as it is 2 cubed, but 9 is not a perfect cube as 9 = 2.08008382...³

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The cube root of this number is 1 more than the smallest prime?

The number 27 has a cube root of 3, which is 2 (the smallest prime) plus 1.

The cube root of this number is one more than the smallest prime

Here is a method: cube root of 400g = n, where n is an integer cube both sides: 400g = n3 then: g = n3/400 therefore: n3/400 must be an integer if this is so, then n3 must be divisible by 400 with no remainder, and n must be => cube root of 400 which is 7.368 bracket the answer by substitution: let n=8, n cubed = 512 no good let n=12, n cubed = 1728 no good let n=20, n cubed = 8000, 8000/400=20 OK No smaller value of n will be divisible by 400 without a remainder, so g=20 is the smallest positive integer that meets the requirement.