## Thursday, June 5, 2014

### Finding Missing Digit - Divisibility Rule, 13

Category: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 14 for

a. 5?3
b. 137?

Solution:

In finding the missing digit, this method is completely different from the previous divisibility by other numbers because we will use the principles of Algebra in solving for the unknown digit.

a. Consider the given number

Since the last digit of a given number is an odd number, then it is not divisible by 14. A number is divisible by 14 if it is both divisible by 2 and 7. All even numbers are divisible by 2. There's nothing that we can do in order to become divisible by 14 since the last digit of a given number is not an even number. You can assign any number to the missing digit but still, the given number will never become divisible by 14.

b. Consider the given number

A number is divisible by 14 if it is both divisible by 2 and 7. In short, an even number that is divisible by 7. We can use 0, 2, 4, 6, and 8 as the last digit of a given number. To test the divisibility of a number by 7, double the last digit and then subtract it to the remaining digits. If the result is a multiple of 7, then the given number is divisible by 7. Let's do this test to all the last digits.

If the last digit is 0, then the given number becomes

Since the result is not a multiple of 7, then the given number is not divisible by 7. 0 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 2, then the given number becomes

Since the result is a multiple of 7, then the given number is divisible by 7. 2 is the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 4, then the given number becomes

Since the result is not a multiple of 7, then the given number is not divisible by 7. 4 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 6, then the given number becomes

Since the result is not a multiple of 7, then the given number is not divisible by 7. 6 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 8, then the given number becomes

Since the result is not a multiple of 7, then the given number is not divisible by 7. 8 is not the last digit of a given number. In this case, only 2 is the last digit of a given number. Therefore, the possible number is only 1372.