Question

suppose that n is a positive integer written in base 10. prove that n is divisible by 3 if and only if the sum of its decimal digits is divisible by 3. (hint: write out the base 10 expansion of n. can one simplify this expression modulo 3?)

Answer 1

Modulo - The result of multiplying the original number by three and the result of multiplying the digits by three must be the same.

What is modulo?

In mathematics, the phrase modulo, which literally means "with respect to a modulus of," is frequently used to claim that two different mathematical objects can be regarded as comparable if their difference is explained by a third component.

Using the Fermat's little theorem:

$x^n-1 =(x-1)(x^{n-1}+x^{n-2}+....+x+1)$

when x=10. You may easily establish the identity by multiplying it term by word. All terms cancel, excluding the first and last ones. Therefore, any power of ten minus one is divisible by nine, and consequently, by three

Now consider a multi-digit natural number, 43617 for example.

$43617= 4*10^4+3*10^3+6*10^2+1*10+7$

$= 4*9999+3*999+6*99+1*9+(4+3+6+1+7)$

Other than the sum of the digits, every term on the right is divisible by 3. The result of multiplying the original number by three and the result of multiplying the digits by three must be the same.

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