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Continued Proportions

Let n=64064...

Question

Let $n = 640640640643$ , without actually computing $n^{2}$ . Prove that $n^{2}$ leave a remainder 1 when divided by 8.

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Solution

Since 640640640000 is a multiple of 8
$643 \equiv 3 (m o d 8)$
$∴ n = 8 k + 3$ for some positive integer k
$∴ n^{2} = 64 k^{2} + 48 k + 9 = 1 + a$ multiple of 8

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