how?

[molten ivy]

if b=3240 then what is the smallest possible whole natural number N where in Nb is a cube of a natural?

[slender oasis]

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[long crown]

Would prime factorisation of b help?

[molten ivy]

idk

[sturdy hazel]

what have you tried then...?

[molten ivy]

nothing really got stuck figuring it out

[sturdy hazel]

well find the prime factorization of b

[molten ivy]

ok

[sturdy hazel]

and recall that (xy)^3=x^3y^3.

so just ensure all the factors of Nb are cubes.

[molten ivy]

okay so

i got the numbers :2 .2.2.3.3.3.3.5

and the end result was 1

what now?

[long crown]

That means 3240=2×2×2×3×3×3×5

[molten ivy]

yea

wait

hold up

i've thought of something and it's either very dumb or reasonably smart

what if we multiply it by 25

now it equals 2x2x2x3x3x3x5x5x5

[long crown]

Yeah

A cube

Now, are you sure 25 is the smallest number to make it a cube?

[molten ivy]

not sure

maybe

i am afterall very dumb

[long crown]

Yeah everyone is dumb

You already have a 5, you need to multiply 25 to complete the cube.
Like, you can't remove it afterall

[molten ivy]

yea

[long crown]

I think you get the intuition

[molten ivy]

sure

okay then

thanks for the help nanihopayega and omegabet

+close

[slender oasis]

✅ Post closed

[gusty halo]

@molten ivy @long crown you had a typo in the meantime

prime factorisation is 3240=2³×3⁴×5¹ and you want all of these exponents to be divisible by 3, which means you need to multiply by 3²×5² or 225

not just 25

[molten ivy]

yea my bad sorry

should i close the post now or?

[gusty halo]

yeah you can close it now

[molten ivy]

okay then

and sorry for the annoyance

+close