Divisibility by 9 증명

Theorem
A number expressed in decimal notation is divisible by 9 iff the sum of its digits is divisible by 9 .

That is:

N=[a 0 a 1 a 2 …a n ] 10 =a 0 +a 1 10+a 2 10 2 +⋯+a n 10 n is divisible by 9

iff:

a 0 +a 1 +…+a n is divisible by 9 .

Corollary

A number expressed in decimal notation is divisible by 3 iff the sum of its digits is divisible by 3 .

That is:

N=[a 0 a 1 a 2 …a n ] 10 =a 0 +a 1 10+a 2 10 2 +⋯+a n 10 n is divisible by 3

iff:

a 0 +a 1 +…+a n is divisible by 3 .

Proof 1

If N is divisible by 9, then

				N	≡		0mod9
		⟺		a 0 +a 1 10+a 2 10 2 +⋯+a n 10 n	≡		0mod9
		⟺		a 0 +a 1 1+a 2 1 2 +⋯+a n 1 n	≡		0mod9			as 10≡1mod9
		⟺		a 0 +a 1 +⋯+a n	≡		0mod9

■
iff=  if and only if.  <----> iff is a convenient shorthand for if and only if.
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