The cube of a 2-digit integer, $\overline{BR} ^3$ is equal to a 5-digit integer, $\overline{BRAIN}$ .

If $B,R,A,I,N$ are all distinct single digit integers, submit your answer as the 2-digit integer $\overline{BR}$ .

The answer is 32.

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From the equation provided we have:

$\overline{BR} \times 1000 + 1000 > \overline{BR} ^ 3 \geq \overline{BR } \times 1000 .$

Since $10 \leq \overline{BR} \leq 100$ , dividing by $\overline{BR}$ , we get

$1000 + 100 \geq 1000 + \frac{1000} { \overline{BR} } > \overline{BR}^2 \geq 1000.$

Hence, $34 > \overline{BR} > 31$ . We now check the cases

Hence, $\overline{BR} = 32$ .