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. 

 

Finding Missing Digit - Divisibility Rule, 10

Category: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 11 for

a. 23?329
b. 39085?

Solution:

a. Consider the given number

To test the divisibility of a number by 11, group the sum of the alternating digits into two groups and then get their difference. If the result is a multiple of 11, then the given number is divisible by 11. Let's do this for the given number as follows

Since -11 is a multiple of 11, then we don't have to add any number so that it becomes a multiple of 11 and hence, 0 is the missing digit. Therefore, the possible number is 230329.

b. Consider the given number

To test the divisibility of a number by 11, group the sum of the alternating digits into two groups and then get their difference. If the result is a multiple of 11, then the given number is divisible by 11. Let's do this for the given number as follows

Since -9 is not a multiple of 11, then we need to add a number so that it becomes a multiple of 11. If you add -2 (add 2 at the second group), then the answer is -11. 2 is the highest digit that we can use. Therefore, the possible number is 390852. 

Finding Missing Digit - Divisibility Rule, 9

Category: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 10 for

a. 45?897
b. 32681? 

Solution:

a. Consider the given number

Since the last digit of the given number is 7, then it is not divisible by 10. A number is divisible by 10 if the last digit is 0. There's nothing that we can do in order to become divisible by 10 since the last digit of a given number is not 0. You can assign any number to the missing digit but still, the given number will never become divisible by 10. 

b. Consider the given number

 
A number is divisible by 10 if the last digit is 0. Since the missing digit is the last digit, then we can assign 0 so that the given number becomes divisible by 10. Therefore, the possible number is 326810 only.

Finding Missing Digit - Divisibility Rule, 8

Category: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 9 for

a. 13?84
b. 2096?

Solution:

a. Consider the given number 

A number is divisible by 9 if the sum of the digits is a multiple of 9. If you add the rest of the digits, the sum will be equal to

Since 16 is not a multiple of 9, then we need to add a number so that it becomes a multiple of 9. So, 16 + 2 = 18. 2 is the highest digit that we can use because 2 + 9 = 11 will be a two digit number and we need to use only one digit to fill up the missing digit. Therefore, the possible number is 13284 only.

b. Consider the given number

A number is divisible by 9 if the sum of the digits is a multiple of 9. If you add the rest of the digits, the sum will be equal to

Since 17 is not a multiple of 9, then we need to add a number so that it becomes a multiple of 9. So, 17 + 1 = 18. 1 is the highest digit that we can use because 1 + 9 = 10 will be a two digit number and we need to use only one digit to fill up the missing digit. Therefore, the possible number is 20961 only.

Finding Missing Digit - Divisibility Rule, 7

Category: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 8 for

a. 78?45
b. 2468?

Solution:

a. Consider the given number

Since the last digit of a given number is not an even number, then the given number is not divisible by 8. The multiples of 8 are all even number. There's nothing that we can do in order to become divisible by 8 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 8. 

b. Consider the given number

A number is divisible by 8 if the last three digit is a multiple of 8. Since the missing digit is the last digit, then we can assign even number digits so that the last three digit will be divisible by 8. If the last digit is 0, then 68 becomes 680 and 680 is divisible by 8. If the last digit is 2, then 68 becomes 682 and 682 is not divisible by 8. If the last digit is 4, then 68 becomes 684 and 684 is not divisible by 8. If the last digit is 6, then 68 becomes 686 and 686 is not divisible by 8. If the last digit is 8, then 68 becomes 688 and 688 is divisible by 8. Therefore, the possible numbers are 24680 and 24688.

Finding Missing Digit - Divisibility Rule, 6

Category: Arithmetic, Algebra

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 7 for

a. 2?4
b. 108?

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 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 for the given number as follows

Next, equate this to the first multiple of 7 which is 7, we have

Since the answer is negative, then we cannot accept this one because we need a positive value for the unknown digit. Let's equate the above equation to the next multiple of 7 which is 14, we have

Since the answer is positive, then we can accept this one. Let's equate the above equation to the next multiple of 7 which is 21, we have

Since 9 is the highest digit, then we can end this process because we want a digit that is less than 10. Therefore, the possible numbers are 224 and 294. You can check these numbers by using a calculator and these numbers are divisible by 7. 

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 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 for the given number as follows

Next, equate this to the multiple of 7 which is close to 108. We want a digit that is positive, whole number, and less than 10. Let's try 98 first, we have

Since the answer is a positive whole number, then we can accept this one. Next, try to equate the above equation to the next multiple of 7 which is 105, we have

Since the answer is not a whole number, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 7 which is 112, we have

Since the answer is a negative number, then we cannot accept this one also. If you will continue this process to the next multiple of 7 like 119, 126, 133 and so on, all values of x are negative. We have to assign a multiple of 7 that is less than 108. Let's try 91 first, we have

Since the answer is not a whole number, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 7 which is 84, we have

Since the answer is greater than 10, then we cannot accept this one also. If you will continue this process to the next multiple of 7 like 77, 70, 63 and so on, all values of x are greater than 10. Therefore, the possible number is 1085 only. You can check this number by using a calculator and this number is divisible by 7.