# Numbers Curiosities

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!

#### The cyclic number: 142857

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)

#### The magic number: 1089

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

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.

#### Three-digit numbers

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!

#### Calculate square powers

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

#### Pairs of squares

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

#### The fantastic number: 12345679

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

#### Reversing and subtracting

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