Divisibility

This post deals with the divisibility of numbers. Specifically, rules that determine if one integer is divisible by another by examination of the number's digits. Prime numbers have no divisors other than one and themselves. This blog has several prior posts on prime numbers: Blogger: Posts.    

Here is a list of rules to apply to determine if a number (greater than 0) is divisible by another:

Divisibility by:

1: All integers are divisible by one so we won't need any help. However, one might ask why we didn't start with zero. Divisibility by zero is undefined. If you use a calculator or a calculator app and divide by zero, you'll get an error message. Therefore, divisibility rules start with the integer 1.

2: All even numbers are divisible by 2. Numbers greater than zero ending in 0, 2, 4, 6 or 8 are even numbers. If the number written in the binary base ends in zero, then the number is divisible by two. See the prior post (Ethiopian Multiplication), for learning how to convert a number to base 2.

3: Sum the digits in the number. If this sum is divisible by 3, the original number will be divisible by 3. If the starting number is very long and you can't determine immediately that the sum of the digits is divisible by 3, apply the rule to sum and continue until you determine the number is or is not divisible by 3. This method will be used for other divisibility rules as well.

4: One hundred and all multiples of 100 are divisible by 4. Therefore, examine the last two digits of the number. If that number formed by the last two digits is divisible by 4, then the original number is divisible by 4.

5: If the last digit is 0 or 5, the number is divisible by 5.

6: Apply the rules for 2 and 3. The number must be even and the sum of its digits are a multiple of three. Another rule (for just some multiples of 6): if the number is greater than 4 is sandwiched by two primes, then the number is a multiple of six (see the prior post: Twin Prime Sandwich).

7: For smaller numbers - drop the last digit (the ones digit) from the number and subtract twice this digit from the remaining number. The original number is a multiple of 7 if the result is a multiple of 7. Example: start with 336. Drop 6, double it and subtract from 33. 33 minus 12 is 21 which is a multiple of 7 so 336 is a multiple of 7.

For larger numbers (>1000) - starting with the right-most digits, alternately sum and subtract each block of three digits. If the result is a multiple of 7, the original number will be a multiple of 7. For example: start with 1,234,567. Sum 567-234+1 = 334 (not a multiple of 7). Try 8,641,969 to get the sum 969-641+8 = 336 which is divisible by 7 (see example above for smaller numbers).

8: Similar to the method for multiples of 4, the number 1000 and all its multiples are divisible by 8. Therefore, if the last 3 digits are a multiple of 8, the original number will be a multiple of 8. Example, start with 2104. The number 104 is divisible by 8 (8x13=104); therefore 2104 is divisible by 8 (8x263=2104). If it is not clear that the number of the last three digits is divisible by 8, divide the number by 2 and now perform the check for divisible by 4 (check to see if the number formed by the last two digits is divisible by 4). Also, as a check for divisibility by 8 is to see if you can divide the number by 2 three times (since 8 = 2 x 2 x 2). For example, 104/2=52, 52/2=26, 26/2=13.

9: Sum the digits in the number. If the resulting sum is divisible by 9, the original number will be divisible by 9. This rule was the basis of the magic trick given in a prior post (Missing Digit Trick).

10: If the last digit  of the number is 0, the number is divisible by 10. 

11: Subtract the last digit (the ones digit) from the remaining digits. If the resulting answer is divisible by 11, then so is the original number. Example: start with 1001. Subtract 1 from 100 to get 99  which is divisible by 11 (11 x 91 = 1001).

12: A number divisible by 12 must meet both rules for divisibility by 3 and 4.

At this point, the reader can see that rules for divisibility of higher compound numbers can be based on combining the divisibility rules of the number's multiples. The more difficult divisibility rules are for higher prime numbers. Wikipedia gives a good summary for numbers up to 30 here.

Credit given to Numberphile for inspiring me to add this post. Numberphile has a good video about divisibility rules and includes more magic tricks based on these rules.