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Nth Term of GP

If 3 2007 -...

Question

If $3^{2007} - 1003$ is divided by $8$ then remainder is

0

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Solution

The correct option is B $0$
$3^{2007} - 1003 = 3 {(9)}^{1003} - 1003 = 3 {(8 + 1)}^{1003} - (125 \times 8 + 3)$

$= 3 [1 + 8 + 8^{2} + . . . . . + 8^{1003}] - 125 \times 8 - 3 = 3 + 8 k - 123 (8) - 3 = 8 [k - 125] = 8 λ$

this implies the given expression is a multiple of $8$

$∴$ Remainder $= 0$

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