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 Mathematics: Interesting facts and numbers

Some beautifull examples with multiplication

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12345679 x 9 = 111111111;

12345679 x 8 = 98765432;

An interesting number 2519
Let us look at 2519 Mod n (n = 2, ... 10).
2519 Mod n means reminder portion of (2519\n), where "\" is the integer division.
 2519 Mod 2 = 1; 2519 Mod 3 = 2; 2519 Mod 4 = 3; 2519 Mod 5 = 4; 2519 Mod 6 = 5; 2519 Mod 7 = 6; 2519 Mod 8 = 7; 2519 Mod 9 = 8; 2519 Mod 10 = 9. 2519 = 1259 x 2 + 1; 2519 = 839 x 3 + 2; 2519 = 629 x 4 + 3; 2519 = 503 x 5 + 4; 2519 = 419 x 6 + 5; 2519 = 359 x 7 + 6; 2519 = 314 x 8 + 7; 2519 = 279 x 9 + 8; 2519 = 251 x 10 + 9.

Example of wrong proof

Find a mistake in the following chain of arguments, pretending to prove that 2=1
 1) Let a = b 2) Multiply 1) by a a2 = ab 3) Add a2 – 2ab to both parts of 2) a2 + a2 – 2ab = ab + a2 – 2ab 4) 3) could be simplified: 2a2 – 2ab = a2 – ab 5) It is the same as 2(a2 – ab) = 1(a2 – ab) 6) Reduce 5) by (a2 – ab). 2=1

Where is a mistake?

Mistake is in the 6th step.
We can not divide by (a2ab) because a2ab = 0.
a = b, so a2ab = 0.

An interesting fact about primes
Mathematicians of XVIIIth century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are primes. It was a big temptation to think that all numbers of such kind are primes. But the next number is not a prime.
333333331 = 17 * 19607843

An elegant proof that
It is obvious that 1 = (2 -1).
= * (2 -1) = (1 + 2 + 22 + ... + 2n) * (2 -1) =
(2 + 22 + 23 ... + 2n+1) - (1 + 2 + 22 + ... + 2n) = 2n+1 - 1.

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