Some
beautifull examples with multiplication
;
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
a^{2}
= ab

3) 
Add
a^{2}
– 2ab
to both parts of 2)
a^{2}
+ a^{2}
– 2ab
= ab
+ a^{2}
– 2ab 
4) 
3)
could be simplified:
2a^{2}
– 2ab
= a^{2}
– ab 
5) 
It
is the same as
2(a^{2}
– ab)
= 1(a^{2}
– ab) 
6) 
Reduce
5) by (a^{2}
– ab).
2=1 
Where is a mistake?
Mistake is in the 6^{th}
step.
We can not divide by
(a^{2}
– ab)
because
a^{2}–
ab
= 0.
a
= b,
so a^{2}–
ab
= 0.
An interesting
fact about primes
Mathematicians of XVIII^{th}
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
+ 2^{2}
+ ... + 2^{n})
*
(2
1)
=
(2 + 2^{2}
+ 2^{3}
... + 2^{n+1})
 (1
+ 2
+ 2^{2}
+ ... + 2^{n})
= 2^{n+1}
 1.
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us if you know
interesting math facts or examples and
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