# A problem concerning divisibility and the number 31. (Number theory)

1. Jul 8, 2014

### TheSodesa

1. The problem statement, all variables and given/known data

Basically, I'm working on a problem where I'm supposed to find a missing digit in a social security number.

The number is as follows: 301 X91 - 2005. where X is the missing digit.

Now, how these numbers are constructed, is that the first six numbers are the persons date of birth, the three numbers that come after those are a personal identification number, and the last symbol is a checking sign, that could be either a number or a letter.

How the last symbol is determined, is that the number formed by the first 9 numbers is divided by 31, and if the remainder is between 0-9, the last symbol becomes the remainder itself, but if the remainder goes above 9, the last symbol becomes a letter, such that 10 = A, 11 = B, 12 = C and so on.

2. Relevant equations and theorems

1) a = bq + r, where a is the dividend, b is the divisor, q is the quotient and r the remainder.

2) If a and b are divisible by c, then the sum a + b is also divisible by c.

3) If a is divisible by c and n is some whole number, then na is also divisible by c.

3. The attempt at a solution

In our case, we can see that the last symbol in the string of numbers is 5, so the remainder must have been 5 when the number 301 X91 200 was divided by 31.

=> 301 X91 200 = 31q + 5 => 31q = 301 X91 200 - 5 = 301 X91 195

=> The number 301 X91 195 must be divisible by 31. 31 is a prime so none of the fancy divisibility rules taught in the US are goign to help me with this. (Not trying to sound condescending.)

And this is where I'm stuck. I'm wondering if turning the number 301 X91 200 into a sum of some form would help me solve this problem based on theorem (2)), but regardless, some insight would be appreciated.

2. Jul 8, 2014

### TheSodesa

I just realized, that X can only be 0, 1 or 2, because it's in the months-position in the birth date -column. As in the 30th of the 10th, 11th or 12th month, 1991. That should narrow down my options.

3. Jul 8, 2014

### TheSodesa

Solved.

Through the process of elimination I calculated that only 301191195 is divisible by 31, out of the 3 options(the other ones being 301091195 and 301291195).

=> X = 1