Numbers Curiosities

This number, if multiplied by 1, 2, 3, 4, 5, 6, 8 or 9, results in another number whose digits are in the same order as the first. If by chance, the result has 7 digits instead of 6, simply add the first with the last digit to get the sequence again. Try it:

`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)`

This number is known to be magical. Look:

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`

Friendly numbers are two numbers where one of them is the sum of the divisors of the other. Watch:

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.

Try to choose any number with three digits and multiply it successively by 7, 11 e 13. See what happens:

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`

Numbers 12 and 21 have the following property: in addition to one being written in reverse order of the other, their squares are also written in reverse order. Look:

`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`

Look what happens if we multiply the number 12345679 by any multiple of 9, between 9 and 81:

`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`

Note that if you subtract any number by itself written in reverse order, the result will always be a multiple of 9:

`21-12=9`

`63-36=27`

`94-49=45`

`311-113=198`