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I am a five digit narcissistic cube who likes to display my root right upfront. I also have hidden it in the back of my four digit cube brother. Who are we?

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$32768 = 32^3$ and $5832 = 18^3$

Process of finding them:

The cube root of a 5 digit number would have to be 2 digits. So say, $AB^3 = ABCDE$. Then, divide by $AB$ to get an approximation $AB^2 \approx 1000$. $\sqrt{1000} \approx 31.6$, so it makes sense to try slightly larger (since our approximation is a clear underestimate). We can try $32^3 = 2^{15} = 32768$ and sure enough it satisfies our condition. ($33^3 = 1089 \cdot 33 > 1100 \cdot 32 = 35200$ can be tested to not work, so this is our five digit cube).
To get the four-digit number, we know that $32$ appears at the end of the cube. The cubes have residues $0, 1, 8, 7, 4, 5, 6, 3, 2, 9 \bmod 10$, so we know that the cube root must be $8 \bmod 10$. Since our four-digit number must have a two-digit cube root, $18^3$ makes the most sense to try ($28^3 > 784 \cdot 20 > 15000$ would be way too large). So therefore we have our answer.

Afternote

Narcissistic numbers have an alternative meaning, so this was a little misleading for awhile.
Perhaps I should justify $1089 \cdot 33 > 1100 \cdot 32$. Well, $1100$ is roughly 1% larger than 1089 (it's 11 more), whereas $32$ is roughly 3% smaller than $33$.

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  • 1
    $\begingroup$ You just beat me to the writeup. :) $\endgroup$ – Rubio May 8 at 20:04
  • $\begingroup$ Alternative proof: 1089*33=(1100-11)*(32+1)=1100*32+1100*1-11*32-11*1 $\endgroup$ – Acccumulation May 9 at 17:52

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