First slide

Binomial theorem for positive integral Index

Question

The remainder when _$7^{103}$ is divided by 25

Difficult

A

0
B

18
C

16
D

9

Solution

$7^{103} = 7 {(49)}^{51} = 7 {(50 - 1)}^{51}$ = $7 (50^{51} - {}^{51}C_{1} 50^{50} + {}^{51}C_{2} 50^{49} - ... - 1)$ $= 7 (50^{51} - {}^{51}C_{1} 50^{50} + {}^{51}C_{2} 50^{49} - ...) + (- 7 - 18 + 18)$ $= 7 (50^{51} - {}^{51}C_{1} 50^{50} + {}^{51}C_{2} 50^{49} - ...) + ((- 25) + 18)$
$∴$ when $7^{103}$ is divided by 25remainder is18

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