bla-bla :: [1] il latino apre la mente - ma non quanto apre il culo!

If you are not familiar with congruences and modular arithmetic, consider this...

Let the number N = w + 10x + 100y + 1000z + ... (example : 574 = 4 + 7*10 + 5*100 )
Its digits are w, x, y, z, ... So the sum of the digits is w + x + y + z + ... = S, say.

So, N - S = D = (w-w) + (10x-x) + (100y-y) + (1000z -z) + ... = 9x + 99y + 999z + ... = 9 (x + 11y + 111z + ...)

So, N - S is a multiple of 9.


Now for the second part...

Consider again, N = w + 10x + 100y + 1000z + ...
We want to prove that, if the sum of its digits, S is divisible by 9, then so is the number, N, and conversely.

If S is divisible by 9, then S = 9p. Now to this, add the number D, calculated above. S + D = 9p + D. But we saw previously that D itself is a multiple of 9, so D = 9q. Hence, S + D = 9p + 9q = 9(p+q) = 9r, say. But then, S + D is nothing but N. So N = 9r, which is what we wnted to prove.

The converse is proved by starting with N = 9r and subtracting D to yield S = 9(r-q) = 9p, say.
QED


mi pare banale che il teorema sia generalizzabile per ogni base. evviva.

" "ciao, io vado a vivere in un appartamento un po' grande e volevo un animale da compagnia: cosa mi consigli?""ti consiglio....l'unicorno!""ma l'unicorno non esiste""non avevi specificato che doveva essere un animale esistente" " (maranza)

ho deciso che vincono le piramidi LOL

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