In arithmetic, divisibility rules are a set of divisibility conditions that apply to a number to determine whether it is divisible by a certain number or not. Basic divisibility rules allow us to determine factors and multiples of numbers without having to divide them by a large number. They are a mental shortcut to finding the answer to complicated divisions.
What is a number divisibility rule?
A number divisibility rule is a trick to determine if a given integer is divisible by a divisor by looking at its digits rather than going through the entire division process. Rather than finding a divisor by trial and error, divisibility tests give you the answer. A number’s divisor is an integer that divides the number fully.
Martin Gardner, a prominent math and science writer, discussed divisibility principles for 2–12 in a 1962 Scientific American article. According to his research, basic divisibility rules were known during the Renaissance and were employed to reduce fractions with high numbers to their simplest terms. Some rules can help us figure out what a number’s true divisor is merely by looking at its digits.
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The divisibility rules
Basic divisibility rules from 2 to 12 will be covered in this section. Because any number is divisible by one, the divisibility rule of one is not required. The following are some basic rules of divisibility:
Divisible by 2
- Number that is even
- A number with the last digit being 0, 2, 4, 6, or 8
Example: 140 ends in a 0 so it is divisible by 2.
Divisible by 3
- The sum of all digits (ex: 42 would be 4 + 2 = 6) is divisible by 3
Example: The sum of 252 (2+5+2) is 9 which is divisible by 3.
Divisible by 4
- The final two digits are divisible by 4
- The number ends in 00
Example: 780 is divisible by 4 because 80 is divisible by 4.
Divisible by 5
- The number ends in a 0 or 5.
Example: 2385 is divisible by 5 as it ends in a 5.
Divisible by 6
- If the integer is divisible by both 2 and 3 then it is divisible by 6
Example: 246 is divisible by both 2 and 3 so it is divisible by 6.
Divisible by 7
- Double the last digit (unit’s digit) in the number and subtract the last digit from the remaining digits. The result should be divisible by 7.
Example: 357 is divisible by 7 because when the final digit, 7, is multiplied by 2 and subtracted from the remaining digits, the result is also divisible by 7.
7 x 2 = 14 and 35-14= 21.
Divisible by 8
- The last three digits are divisible by 8
- The number ends in 000
Example: 53,905,256 is divisible by 8 because 256 is.
Divisible by 9
- The sum of the numbers should equal 9, or a multiple of 9
Example: 81 is divisible by 9 because 8+1 = 9. Likewise 711 is divisible by 9 because 7+1+1 = 9. Also, 51984 is divisible by 9 because 5+1+9+8+4= 27 is a multiple of 9.
Divisible by 10
- Any number that ends in a 0
Example: Any number that ends in a 0 is divisible by 10.
Divisible by 11
- The difference of the sum of the digits is 0 or divisible by 11.
Example: This one is more complex than the rest. Take the number 2782, and work out the difference between the sums: 2-7 + 2-8 = -11 which is divisible by 11.
Divisible by 12
- A number that is divisible by 4 and 3.
Example: If the number is divisible by both 3 and 4 then it is also divisible by 12.
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Factors can be useful
Factors can be useful in mathematics, since they are numbers or algebraic expressions that divide another number or expression evenly, meaning there is no remainder. To calculate the factors of a number, we must first determine which numbers can be divided into it. Here’s where the rules of divisibility come in handy!
For example, let’s find the factors of the number 230. We can use the divisibility rules to find the factors of this number. As it ends in a 0, we know that it’s divisible by 10 and 5. As such, we can divide 230 by both of these to find its factor pairs:
230 / 10 = 23
230 / 5 = 46
We also know that it is divisible by 2 as it can be halved without creating a decimal:
230 / 2 = 115
Does it fit any other divisibility rules? No, therefore the factors of 230 are: 1, 2, 5, 10, 23, 46, 115, and 230.
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Divisibility for Prime Numbers
Prime numbers that are less than 20 and bigger than 10 are divided using intermediate divisibility criteria. The prime numbers 2, 3, 5, 7, and 11 have already been tested for divisibility.
Rules of 13
When we divide a number by 13 and the rest of the number is 0, we call it divisible by 13. Without completing a lengthy division, the divisibility test of 13 allows us to rapidly determine whether a number is divisible by 13. First, we must multiply the unit digit by 4 according to the divisibility rule of 13. Then, excluding the digit at the unit’s location, we add the product to the remainder of the number to its left. If the result is a number divisible by 13, the original integer is divisible by 13 as well.
Rules of 17
When 17 divides an integer fully, it is said to be divisible by 17. First, we must multiply the unit digit by 5 according to the divisibility rule of 17. Then, excluding the digit at the unit’s location, we subtract the product from the remainder of the number to its left. If the difference produces a number that is divisible by 17, the original number is also divisible by 17.
Rules of 19
When we divide a number by 19 and receive 0 as the remainder, the number is said to be divisible by 19. First, we must multiply the units digit by 2 according to the divisibility rule of 19. Then, excluding the digit at the unit’s location, we add the product to the remainder of the number to its left. If the result is a number divisible by 19, the original number is divisible by 19 as well.
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