# Powers of 2 and 5 without Zeroes

## Theorem

The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:

$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$

## Proof

$n$ $2^n$ $5^n$
$0$ $1$ $1$
$1$ $2$ $5$
$2$ $4$ $25$
$3$ $8$ $125$
$4$ $16$ $625$
$5$ $32$ $3125$
$6$ $64$ $15 \, 625$
$7$ $128$ $78 \, 125$
$9$ $512$ $1 \, 953 \, 125$
$18$ $262 \, 144$ $3 \, 814 \, 697 \, 265 \, 625$
$33$ $8 \, 589 \, 934 \, 592$ $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$

It is probable that $n = 33$ is the final instance.

$\blacksquare$