2. In the first chapter of Disquisitiones Arithmeticae, Gauss introduced the con

Question

2. In the first chapter of Disquisitiones Arithmeticae, Gauss introduced the concept of congruence. He explained that he was induced to adopt the symbol is congruent to because of the close analogy with algebraic equality. Here is Gauss' definition: we define a is congruent to b (mod n) if n divides the difference a - b; in other words, a - b = kn for some integer k. Now, any positive integer n can be written in decimal form. n = ak10^k + ak-110^k-1 + ? + a110 + a0 where 0 less than or equal to aj less than or equal to 9 for all j. Prove that (a) n is congruent to a0 + a1 +? + ak(mod 9). Consequently, 9 divides n if and only if 9 divides the sum of the digits aj (b) n is congruent to a0 ? a1 + a2 - ? + (-1)^kak(mod 11). Consequently 11 divides n if 11 divides the alternating sum of the digits aj.

Explanation / Answer

The number n is
n = 10kak + 10k-1ak-1 + . . . + 101a1 + a0.
Since 10 ? 1 mod 9, it follows that 10k ? 1 mod 9. So
n ? 10kak + 10k-1ak-1 + . . . + 101a1 + a0 ? ak +

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