Using Binomial theorem, find the remainder when $5^{103}$ is divided by $13$ .

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Solution

$5^{103}$
$= 5 (26 - 1)^{51}$
$= 5 (. . . . . . . . . . . . .^{51} C_{51} (- 1)^{51})$
$↓$
All other terms coming
in expansion will be
divisible by $13$ since
they have $26$ as a
factor
$= - 5$
But remainder is always positive
So, $(- 5 + 11) = 8 \to r e m$ .

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