Iii the first chapter of Disquisitiones Arithmeticae. Gauss introduced the conce
ID: 3081411 • Letter: I
Question
Iii the first chapter of Disquisitiones Arithmeticae. Gauss introduced the concept of congruence. Here is Gauss' definition: we define a = 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 = ak 10k + a k-1 + 10 k-1 +...+ a1 10 + a0 where 0 le aj le9 for all j. Prove that n = ao + a1+...+ ak(mod 9). Consequently, 9 divides n if and only 9 divides the sum of the digits aj. n = ao - a1+ a2 -... +(-1)kak (mod 11). Consequently 11 divides n if 11 divides the alternating sum of the digits aj.Explanation / Answer
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