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.

Finding Missing Digit - Divisibility Rule, 12

Category: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 13 for

a. 6?6
b. 205?

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

Let x be the unknown ten's digit. The given number can written as

To test the divisibility of a number by 13, multiply the last digit by 4 and then add it to the remaining digits. If the result is a multiple of 13, then the given number is divisible by 13. Let's do this for the given number as follows

Next, equate this to the first multiple of 13 which is 13, we have 

Since the answer is negative, then we cannot accept this one because we need a positive value for the unknown digit. The multiples of 13 are 13, 26, 39, 52, 65, 78, 91, 104, and so on. We need to choose a number which is greater than 84 that is 91 in order to get a positive value of x. Let's equate the above equation to 91, we have

Since the answer is positive, then we can accept this one. We need to end this process because we want a digit that is less than 10. Therefore, the possible number is only 676. You can check this number by using a calculator and this number is divisible by 13.

b. Consider the given number

Let x be the unknown one's digit. The given number can written as 

To test the divisibility of a number by 13, multiply the last digit by 4 and then add it to the remaining digits. If the result is a multiple of 13, then the given number is divisible by 13. Let's do this for the given number as follows

  

Next, equate this to the multiple of 13 which is close to 205. We want a digit that is positive, whole number, and less than 10. Let's try 208 first, we have 

Since the answer is a fraction, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 13 which is 221, we have 

Since the answer is positive, then we can accept this one. We need to end this process because we want a digit that is less than 10. Therefore, the possible number is only 2054. You can check this number by using a calculator and this number is divisible by 13.
 

Finding Missing Digit - Divisibility Rule, 11

Category: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 12 for

a. 234?89
b. 34524?

Solution:

a. Consider the given number 

A number is divisible by 12 if it is both divisible by 3 and 4. Since the last digit of a given number is an odd number, then the given number is not divisible by 12. The multiples of 12 are all even number. There's nothing that we can do in order to become divisible by 12 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 12.  

b. Consider the given number

A number is divisible by 12 if it is both divisible by 3 and 4. Since the missing digit is the last digit, then we can assign even number digits so that the last two digit becomes divisible by 4. When you add all the digits, the sum should be a multiple of 3 so that the given number is divisible by 12. If the last digit is 0, then the last two digit becomes 40. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 0 = 18. Since 18 is a multiple of 3, then 0 is the last digit. If the last digit is 4, then the last two digit becomes 44. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 4 = 22. Since 22 is not a multiple of 3, then we cannot use 4 as the last digit. If the last digit is 8, then the last two digit becomes 48. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 8 = 26. Since 26 is not a multiple of 3, then we cannot use also 8 as the last digit. Therefore, the possible number is 345240 only.