Follow

follow

Dear Michel Planat,

The Symmetry in number theory includes deep problems, some of them related to the distribution of primes and the Riemann hypothesis.

Here, I consider an elementary property that connects any natural number with its symmetry, just for fun.

Consider any positive integer number,at least two of its digits are distinct

N=a₁a₂a₃....an,the symmetric of this number is defined as

S(N)=anan-1an-2....a₁

We can prove the following nice (elementary) property:

The digit 9 divides |N-S(N)|.

Using this property one can use it to form a beautiful puzzle for the secondary school students as follows:

(1) Guess any number with any number of digits at least 2 of them are distinct.

(2) Reverse the order of the digits to obtain a symmetric number to the original one.

(3) Ask the student to find the difference between them. ( The greatest minus the smallest).

(4) Ask him to hide any digit of the answer and to announce the remaining digits.

Now you can tell the student what the missing digit ( the hidden one) is.

The answer chooses the digit that makes the announced number divisible by 9.

Example:

Assume the student guesses the number N=745931

It's symmetric is S(N)=139547

Now, N-S(N)=745931-139547= 606384.

The student will hide any digit, say 6.

Then he announces the following:60384.

How can I find the hidden digit?

Using the fact that N-S(N) is divisible by nine and the sum of the digits should divide the nine. The sum of the digits of the announced number is

6+0+3+8+4=21, Not divisible by 9.

So we need to add 6 making 21+6=27 divisible by 9. Therefore the hidden

number is 6.

PS. If N-S(N) is divisible by 9, the unknown number is 0 or 9.

Best regards

Thank you Issam. It is numerology at a good level. But probably insufficient to do as well as the RSA algorithm based on the factorization of integers.