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

In summary, the problem involves finding a missing digit in a social security number, where the last symbol is determined by dividing the first 9 numbers by 31. The possible options for the missing digit are narrowed down to 0, 1, or 2, and through the process of elimination it is determined that the missing digit is 1. The summary also includes relevant equations and theorems used to solve the problem.
  • #1
TheSodesa
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7

Homework Statement



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. Homework 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.

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.
 
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  • #2
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
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
 

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

1. What is the problem concerning divisibility and the number 31 in number theory?

The problem is to determine the divisibility of a given number by 31. In other words, can the given number be divided evenly by 31 without any remainder?

2. Why is the number 31 significant in this problem?

The number 31 is significant because it is a prime number. Prime numbers are numbers that can only be divided by 1 and themselves. In this case, 31 cannot be divided by any other number besides 1 and 31, making it an important number in number theory.

3. How can I test if a number is divisible by 31?

One way to test if a number is divisible by 31 is by using the divisibility rule for 31: subtract three times the last digit of the number from the remaining digits. If the result is divisible by 31, then the original number is also divisible by 31.

4. What are some examples of numbers that are divisible by 31?

Some examples of numbers that are divisible by 31 are 62, 93, 124, 155, and 186. These numbers can be divided evenly by 31 without any remainder.

5. Is there a mathematical formula for determining divisibility by 31?

Yes, there is a formula for determining divisibility by 31. It is: (3n+1)/10, where n is the remaining digits after removing the last digit of the number. If the result is a whole number, then the original number is divisible by 31.

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