# Amicable Pair/Examples/10,744-10,856

## Example of Amicable Pair

$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:

$\sigma \left({10 \, 744}\right) = \sigma \left({10 \, 856}\right) = 21 \, 600 = 10 \, 744 + 10 \, 856$

## Proof

By definition, $m$ and $n$ form an amicable pair if and only if:

$\sigma \left({m}\right) = \sigma \left({n}\right) = m + n$

where $\sigma \left({n}\right)$ denotes the $\sigma$ function.

Thus:

 $\displaystyle \sigma \left({10 \, 744}\right)$ $=$ $\displaystyle 21 \, 600$ $\sigma$ of $10 \, 744$ $\displaystyle$ $=$ $\displaystyle 10 \, 744 + 10 \, 856$ $\displaystyle$ $=$ $\displaystyle \sigma \left({10 \, 856}\right)$ $\sigma$ of $10 \, 856$

$\blacksquare$