Finding Missing Digit - Divisibility Rule

Category: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 2 for

a. 245?7
b. 3259?

Solution:

a. Consider the given number

Since the last digit of a given number which is 7 is an odd number, then the given number is not divisible by 2. There's nothing that we can do in order to become divisible by 2 since the last digit of a given number is an odd number. You can assign any number to the missing digit but still, the given number will never become divisible by 2. 

b. Consider the given number

The given number is divisible by 2 if the last digit is an even number. Since the missing digit is the last digit, then we can assign 0, 2, 4, 6, and 8 in order to become divisible by 2. Therefore, the possible numbers are 32590, 32592, 32594, 32596, and 32598.

Divisibility Test - Numbers, 2

Category: Arithmetic

"Published in Vacaville, California, USA"

Perform the divisibility test of a number by 1 to 15 for 

Solution:

Consider the given number above 

Without using a calculator, we can perform the divisibility test by using divisibility rules.

Divisibility by 1:

All numbers are divisible by 1 because 1 is a universal factor. Any number divided by 1 is always the same number.

Divisibility by 2:

Since the given number is an odd number, then it is not divisible by 2. All even numbers are divisible by 2. 

Divisibility by 3:

To test a given number, add all the digits of a given number as follows 

Since 6 is a multiple of 3, then the given number is divisible by 3.

Divisibility by 4:

Consider the given number  

Since the last two digits of a given number is an odd number, then the given number is not divisible by 4. The multiples of 4 are all even numbers.

Divisibility by 5:

Since the last digit of a given number is 5, then the given number is divisible by 5. A number that ends with 5 or 0 is divisible by 5.  

Divisibility by 6:

Since the given number is an odd number, then it is not divisible by 6. If an even number that is divisible by 3, then it is divisible by 6.

Divisibility by 7:

To test a given number, double the last digit and then subtract it to the rest of the digits as follows 

   

Since the result of this process is 0, then the given number is divisible by 7. If the result of the process is a multiple of 7 or 0, then the given number is divisible by 7.

Divisibility by 8:

Consider the given number 

Since the last three digits of a given number is an odd number, then the given number is not divisible by 8. The multiples of 8 are all even numbers. 

Divisibility by 9:

To test a given number, add all the digits of a given number as follows  

Since 6 is not a multiple of 9, then the given number is not divisible by 9. 

Divisibility by 10:

Since the last digit of a given number ends with 5, then the given number is not divisible by 10. A number that ends with 0 is divisible by 10.  

Divisibility by 11:

To test a given number, create two groups of the sum of alternating digits and then get their difference as follows

   

Since -1 is not a multiple of 11, then the given number is not divisible by 11.

Divisibility by 12:

Since the given number is divisible by 3 only and not divisible by 4 because it is an odd number, then the given number is not divisible by 12. 

Divisibility by 13:

To test a given number, multiply the last digit by 4 and then add it to the rest of the digits as follows

 

 

 

Since 39 is a multiple of 13, then the given number is  divisible by 13.

Divisibility by 14:

Although the given number is divisible by 7 but it is not divisible by 2 because it is an odd number. Because of this, the given number is not divisible by 14.
 
Divisibility by 15:

Since the given number is both divisible by 3 and 5, then it is divisible by 15. 

Therefore,

1365 is divisible by 1, 3, 5, 7, 13, and 15. 

Divisibility Test - Numbers

Category: Arithmetic

"Published in Newark, California, USA"

Perform the divisibility test of a number by 1 to 15 for 

Solution:

Consider the given number above

Without using a calculator, we can perform the divisibility test by using divisibility rules.

Divisibility by 1:

All numbers are divisible by 1 because 1 is a universal factor. Any number divided by 1 is always the same number.

Divisibility by 2:

Since the given number is an even number, then it is divisible by 2. All even numbers are divisible by 2.

Divisibility by 3:

To test a given number, add all the digits of a given number as follows

   
Since 9 is a multiple of 3, then the given number is divisible by 3.

Divisibility by 4:

Consider the given number 

Since 80 which is the last two digits of a given number is a multiple of 4, then the given number is divisible by 4.

Divisibility by 5:

Since the last digit of a given number is 0, then the given number is divisible by 5. A number that ends with 5 or 0 is divisible by 5. 

Divisibility by 6:

Since the given number is both divisible by 2 and 3, then it is divisible by 6.

Divisibility by 7:

To test a given number, double the last digit and then subtract it to the rest of the digits as follows
 

 

 

   
You can repeat the process as follows
 

 

 

Since -6 is not a multiple of 7, then the given number is not divisible by 7.

Divisibility by 8:

Consider the given number

Since 080 which is the last three digits of a given number is a multiple of 8, then the given number is divisible by 8.

Divisibility by 9:

To test a given number, add all the digits of a given number as follows 

Since 9 is a multiple of 9, then the given number is divisible by 9.

Divisibility by 10:

Since the last digit of a given number ends with 0, then the given number is divisible by 10. A number that ends with 0 is divisible by 10. 

Divisibility by 11:

To test a given number, create two groups of the sum of alternating digits and then get their difference as follows

Since 9 is not a multiple of 11, then the given number is not divisible by 11.

Divisibility by 12:

Since the given number is both divisible by 3 and 4, then it is divisible by 12. 

Divisibility by 13:

To test a given number, multiply the last digit by 4 and then add it to the rest of the digits as follows 

   

You can repeat the process as follows

  

Since 42 is not a multiple of 13, then the given number is not divisible by 13. 

Divisibility by 14:

Although the given number is an even number but it is not divisible by 7. Because of this, the given number is not divisible by 14.

Divisibility by 15:

Since the given number is both divisible by 3 and 5, then it is divisible by 15. 

Therefore,

1080 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 15.