| 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
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
(a2
– ab)
because
a2–
ab
= 0.
a
= b,
so a2–
ab
= 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.
Contact
us if you know
interesting math facts or examples and
would want to share them. |
