# 217

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## Number

$217$ (two hundred and seventeen) is:

$7 \times 31$

The $3$rd Fermat pseudoprime to base $5$ after $4$, $124$:
$5^{217} \equiv 5 \pmod {217}$

The $9$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$:
$217 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 = 9^3 - 8^3$

The $13$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$ whose divisor sum is square:
$\map {\sigma_1} {217} = 256 = 16^2$

The $43$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.