Here you can find some curiosities about numbers. Although most of them do not have any use for your studies, they are at least, interesting facts for those who like to play with the numbers. Have fun!

`142857 xx 5 = 714285`

`142857 xx 8 = 1142856 text(, adding the the first digit with the last ) (1 + 6) = 7 => 142857`

Even more curious. It is not necessary to use the number 142857, in this order. We can use any order as long as the 6 figures are used. Look:

`428571 xx 2 = 857142`

`285714 xx 3 = 857142`

`285714 xx 9 = 2571426 text(, adding the the first digit with the last ) (2 + 6) = 8 => 571428`

And what happens if it is multiplied by 7?

Well, in that case, we will get a sequence with only the 9. If we get more than 6 digits, those who are not 9, can be summed to give 9.

`142857 xx 7 = 999999`

`857142 xx 7 = 5999994 (5 + 4 = 9)`

Choose any number with three different digits: for example, 254.

Now write this number inverted and subtract the smallest from the largest:

`452 - 254 = 198`

Now add this number to itself, but inverted:

`198 + 891 = 1089`

The result is always the magic number 1089.

Always works as long as 3 digits are used in the calculation.

Another example, I choose the number 574:

`574 - 475 = 099`

`099 + 990 = 1089`

For example, divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 e 110, whose sum is 284.

On the other hand, the divisors of 284 are 1, 2, 4, 71 e 142 and the sum of them is 220

**Fermat** also discovered the pair 17.296 e 18.416.

**Descartes** discovered the pair 9.363.584 e 9.437.056.

For example, the number 361.

`361 xx 7 = 2527`

`2527 xx 11 = 27797`

`27797 xx 13 = 361361`

That is, the first number chosen always appear repeated twice!

**Pythagoras** found that `n^2` is equal to the sum of the first `n` odd natural numbers. Look:

`3^2=1+3+5=9`

`4^2=1+3+5+7=16`

`5^2=1+3+5+7+9=25`

`12^2=144 ^^ 21^2=441`

Other pairs of numbers with the same property:

`13^2=169 ^^ 31^2=961`

`122^2=14884 ^^ 221^2=48841`

The mathematician **Thébault** investigated the pairs that have this curious property and found the following pair:

`1113^2=1238769 ^^ 3111^2=9678321`

`12345679 xx 9 = 111.111.111`

`12345679 xx 18 = 222.222.222`

`12345679 xx 27 = 333.333.333`

`12345679 xx 36 = 444.444.444`

`12345679 xx 45 = 555.555.555`

`12345679 xx 54 = 666.666.666`

`12345679 xx 63 = 777.777.777`

`12345679 xx 72 = 888.888.888`

`12345679 xx 81 = 999.999.999`

`21-12=9`

`63-36=27`

`94-49=45`

`311-113=198`