Divisibility rules are a set of shortcuts that can help you quickly determine whether a number is divisible by another number. They are a great way to learn math in a fun and easy way.
What is the divisibility rule?
A divisibility rule is a trick for determining whether 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 through trial and error, divisibility criteria gives you the answer. The divisor of a number is an integer that completely divides that number.For example, the divisibility rule for 2 states that any number whose last digit is 0, 2, 4, 6, or 8 is divisible by 2.
Martin Gardner, a math and science writer, discussed the principles of divisibility for numbers 2 through 12 in a 1962 article in Scientific American. According to his research, basic divisibility rules were known during the Renaissance and were used to reduce fractions with large numbers to their simplest expression. Certain rules can help us determine the true divisor of a number by simply looking at its digits.
Why are divisibility rules important?
Divisibility rules are important because they can help you save time and energy when working with numbers. For example, if you want to know whether a number is divisible by 5, you don’t have to divide it by 5. Instead, you can simply look at the last digit and see if it is a 0 or a 5. If it is, then the number is divisible by 5.
How to use divisibility rules
To use a divisibility rule, simply check whether the number satisfies the condition specified in the rule. If it does, then the number is divisible by the number specified in the rule.
Divisibility rule for 2
Any number whose last digit is 0, 2, 4, 6, or 8 is divisible by 2.
This rule can be used to quickly determine whether a number is even or odd, and it can also be used to simplify division problems. For example, to divide a number by 2, you can simply divide the last digit by 2. If the last digit is divisible by 2, then the entire number is also divisible by 2.
The divisibility rule of 2 is also useful for solving more complex problems, such as those involving factoring polynomials or finding greatest common factors. By understanding and applying this rule, mathematicians and scientists can save time and effort in solving a wide range of problems.
- Financial analysts: The divisibility rule of 2 can be used to quickly determine whether a number is even or odd, which can be useful for calculating interest rates and other financial metrics.
- Software engineers: The divisibility rule of 2 can be used to develop efficient algorithms and to optimize code.
- Scientists: The divisibility rule of 2 can be used to analyze data and to develop models.
The divisibility rule of 2 is a valuable tool for anyone who uses mathematics and science in their professional life. By understanding and applying this rule, you can save time and effort, and you can also develop more efficient and accurate solutions to problems.
Examples
24 – This number ends in 4, which is an even digit. Therefore, 24 is divisible by 2.
137 – This number ends in 7, which is an odd digit. Therefore, 137 is not divisible by 2.
50 – This number ends in 0, which is an even digit. Therefore, 50 is divisible by 2.
9876 – This number ends in 6, which is an even digit. Therefore, 9876 is divisible by 2.
333 – This number ends in 3, which is an odd digit. Therefore, 333 is not divisible by 2.
Divisibility rule for 3
The sum of the digits of a number is divisible by 3 if and only if the number is divisible by 3.
The divisibility rule of 3 can be used in a variety of professional settings, such as:
- Accounting and finance: The divisibility rule of 3 can be used to quickly and easily check the accuracy of financial calculations. For example, you can use the divisibility rule of 3 to check the accuracy of a tip amount or the total cost of a purchase.
- Data analysis: The divisibility rule of 3 can be used to identify patterns and trends in data. For example, you can use the divisibility rule of 3 to identify customers who are most likely to make a purchase or to identify products that are most likely to sell.
- Software development: The divisibility rule of 3 can be used to develop efficient and accurate algorithms. For example, you can use the divisibility rule of 3 to develop an algorithm for calculating the greatest common divisor of two numbers.
To use the divisibility rule of 3, simply add up the digits of the number. If the sum of the digits is divisible by 3, then the number is also divisible by 3.
Examples
369. To check if it’s divisible by 3, add up its digits: 3 + 6 + 9 = 18. Now, check if 18 is divisible by 3. Since 18 ÷ 3 = 6, which is a whole number, 369 is divisible by 3.
1,782. Add its digits: 1 + 7 + 8 + 2 = 18. Since 18 is also a multiple of 3 (18 = 3 x 6), we can say that 1,782 is divisible by 3.
7,254. Add its digits: 7 + 2 + 5 + 4 = 18. Check if 18 is divisible by 3. 18 ÷ 3 = 6, so 7,254 is divisible by 3.
1,234,567. Sum its digits: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. Check if 28 is divisible by 3. 28 ÷ 3 = 9 with a remainder of 1. Since there’s a remainder, 1,234,567 is not divisible by 3.
Divisibility rule for 4
The last two digits of a number are divisible by 4 if and only if the number is divisible by 4.
The divisibility rule for 4 can be used for a variety of tasks, such as:
- Checking whether a number is divisible by 4 for mathematical calculations
- Determining whether a number is divisible by 4 for financial purposes, such as calculating interest
- Checking whether a number is divisible by 4 for computer science purposes, such as generating random numbers
The divisibility rule for 4 is a simple and effective way to determine whether a number is divisible by 4. It is a useful skill for a variety of tasks, both mathematical and non-mathematical.
Examples
812.The last two digits are 12, and 12 is divisible by 4 (12 ÷ 4 = 3).
Therefore, 812 is divisible by 4.
2,346.The last two digits are 46, and 46 is divisible by 4 (46 ÷ 4 = 11.5, a whole number).
So, 2,346 is divisible by 4.
5,721.The last two digits are 21, and 21 is not divisible by 4 (21 ÷ 4 = 5.25, not a whole number).
Therefore, 5,721 is not divisible by 4.
1,200.The last two digits are 00, and any number ending in 00 is divisible by 4.
So, 1,200 is divisible by 4.
Divisibility rule for 5
Any number whose last digit is 0 or 5 is divisible by 5.
The divisibility rule for 5 is based on the following mathematical principle:
Any number that is a multiple of 5 can be written in the form 5n, where n is an integer.
The last digit of a number in the form 5n is always a 0 or a 5. This is because the product of any integer and the number 5 always has a last digit of 0 or 5.
The divisibility rule for 5 has a variety of applications in mathematics and other fields. For example, it can be used to:
- Quickly check to see if a number is a multiple of 5
- Calculate percentages
- Determine the number of digits in a decimal number
- Convert between different number systems
- Simplify mathematical expressions
- Solve mathematical problems more efficiently
Examples
85
In this case, the units digit is 5, so 85 is divisible by 5.
420
The units digit in 420 is 0, so it is divisible by 5.
267
The units digit here is 7, which is not 0 or 5, so 267 is not divisible by 5.
150
The units digit in 150 is 0, so it is divisible by 5.
3,295
The units digit is 5, so 3,295 is divisible by 5.
Divisibility rule for 6
A number is divisible by 6 if and only if it is divisible by both 2 and 3.
To use the divisibility rule for 6, simply follow these steps:
- Check if the number has an even last digit. If the number has an odd last digit, then it is not divisible by 6.
- Calculate the sum of the digits in the number. If the sum of the digits is not divisible by 3, then the number is not divisible by 6.
Examples
72
To check if 72 is divisible by 6, we can apply the divisibility rule for 6, which states that a number is divisible by 6 if it is divisible by both 2 and 3.
72 is even (divisible by 2) because its units digit is 2.
The sum of the digits of 72 is 7 + 2 = 9, which is divisible by 3.
Since 72 is divisible by both 2 and 3, it is divisible by 6.
150
150 is even (divisible by 2) because its units digit is 0.
The sum of the digits of 150 is 1 + 5 + 0 = 6, which is divisible by 3.
Since 150 is divisible by both 2 and 3, it is divisible by 6.
27
27 is not even (it doesn’t end in 0, 2, 4, 6, or 8), so it’s not divisible by 2.
The sum of the digits of 27 is 2 + 7 = 9, which is divisible by 3.
Since 27 is not divisible by 2, it is not divisible by 6.
13
13 has an odd last digit (3).
The sum of the digits in 13 is 1 + 3 = 4, which is not divisible by 3.
Therefore, 13 is not divisible by 6.
Divisibility rule for 7
To determine whether a number is divisible by 7, double the last digit and subtract it from the remaining number. If the result is divisible by 7, then the original number is also divisible by 7.
To use the divisibility rule for 7, simply follow these steps:
- Double the last digit of the number.
- Subtract the doubled digit from the remaining number.
- If the result is divisible by 7, then the original number is also divisible by 7.
Examples
532
- Take the last digit, which is 2.
- Remove it from the number, leaving 53.
- Double the removed digit (2 * 2 = 4).
- Subtract this doubled digit from the remaining number (53 – 4 = 49).
Since 49 is divisible by 7 (7 * 7 = 49), then 532 is also divisible by 7.
7289
- Take the last digit, which is 9.
- Remove it from the number, leaving 728.
- Double the removed digit (9 * 2 = 18).
- Subtract this doubled digit from the remaining number (728 – 18 = 710).
Now, you have 710. Continue the process:
- Take the last digit, which is 0.
- Remove it from the number, leaving 71.
- Double the removed digit (0 * 2 = 0).
- Subtract this doubled digit from the remaining number (71 – 0 = 71).
71 is not divisible by 7, then 7289 is not divisible by 7.
763
- Double the last digit (3 * 2 = 6).
- Subtract the result from the remaining part of the number without the last digit (76 – 6 = 70).
If the result (in this case, 70) is divisible by 7, then the original number (763) is also divisible by 7. In this case, 70 is divisible by 7, so 763 is divisible by 7.
189
- Take the last digit, which is 9.
- Remove it from the number, leaving 18.
- Double the removed digit (9 * 2 = 18).
- Subtract this doubled digit from the remaining number (18 – 18 = 0).
Since 0 is divisible by 7 (0 * 7 = 0), then 189 is also divisible by 7.
Divisibility rule for 8
The last three digits of a number are divisible by 8 if and only if the number is divisible by 8.
The divisibility rule for 8 can be used in a variety of applications, including:
- Validating credit card numbers
- Identifying multiples of 8
- Calculating the size of a pizza
- Converting between different units of measurement
Examples
4,992. Focus on the last three digits, which are 992. Since 992 is divisible by 8 (8 times 124), the whole number 4,992 is divisible by 8.
1,216. Look at the last three digits, which are 216. Since 216 is divisible by 8 (8 times 27), the entire number 1,216 is divisible by 8.
9,024. the last three digits are 024. Since 24 is divisible by 8 (8 times 3), the entire number 9,024 is divisible by 8.
7,456. Since 456 is divisible by 8 (8 times 57), the whole number 7,456 is divisible by 8.
6,864. the last three digits, which are 864. Since 864 is divisible by 8 (8 times 108), the entire number 6,864 is divisible by 8.
3,760. the last three digits, which are 760. Since 760 is divisible by 8 (8 times 95), the whole number 3,760 is divisible by 8.
Divisibility rule for 9
The sum of the digits of a number is divisible by 9 if and only if the number is divisible by 9.
Examples
243
Sum of Digits: 2 + 4 + 3 = 9
Since the sum of the digits (9) is divisible by 9, the number 243 is divisible by 9.
1,035
Sum of Digits: 1 + 0 + 3 + 5 = 9
(9) is divisible by 9, the number 1,035 is divisible by 9.
72,810
Sum of Digits: 7 + 2 + 8 + 1 + 0 = 18
(18) is also divisible by 9, the number 72,810 is divisible by 9.
4,567,893
Sum of Digits: 4 + 5 + 6 + 7 + 8 + 9 + 3 = 42
Although the sum of the digits (42) is not directly divisible by 9, we can repeat the process: 4 + 2 = 6. Now, 6 is not divisible by 9, so the original number, 4,567,893, is not divisible by 9.
Divisibility rule for 10
Any number whose last digit is 0 is divisible by 10.
Examples
10 is divisible by 10 because it ends with a 0
20 is divisible by 10 because it ends with a 0.
50 is divisible by 10 because it ends with a 0.
100 is divisible by 10 because it ends with a 0.
1,000 is divisible by 10 because it ends with a 0.
2,340 is divisible by 10 because it ends with a 0.
15 is not divisible by 10 because it does not end with a 0.
73 is not divisible by 10 because it does not end with a 0.
987 is not divisible by 10 because it does not end with a 0.
6,789 is not divisible by 10 because it does not end with a 0.
Divisibility rule for 11
The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of its digits at odd positions (1st, 3rd, 5th, etc.) and the sum of its digits at even positions (2nd, 4th, 6th, etc.) is either 0 or a multiple of 11.
Examples
363
The sum of the digits at odd positions is 3 + 3 = 6, and the sum of the digits at even positions is 6. The difference is 6 – 6 = 0. Since the difference is 0, 363 is divisible by 11.
1,210
The sum of the digits at odd positions is 1 + 0 = 1, and the sum of the digits at even positions is 2 + 1 = 3. The difference is 1 – 3 = -2. While the difference is not 0, it’s still divisible by 11 because -2 is a multiple of 11 (since -2 = -11 * 0 + 11 * (-1)). So, 1,210 is divisible by 11.
7,845
The sum of the digits at odd positions is 7 + 4 = 11, and the sum of the digits at even positions is 8 + 5 = 13. The difference is 11 – 13 = -2, which is not 0. 7,845 is divisible by 11.
2,310
The sum of the digits at odd positions is 2 + 1 = 3, and the sum of the digits at even positions is 3 + 0 = 3. The difference is 3 – 3 = 0. Therefore, 2,310 is divisible by 11.
2782
Sum the digits at odd positions (1st and 3rd digits): 2 + 8 = 10.
Sum the digits at even positions (2nd and 4th digits): 7 + 2 = 9.
Find the difference between these sums: 10 – 9 = 1.
In this case, the difference is 1, which is not equal to 0, and it’s not a multiple of 11. Therefore, based on the divisibility rule for 11, the number 2782 is not divisible by 11.
Divisibility rule for 12
a number is divisible by 12 if and only if it is divisible by both 3 and 4. This is because 12 is the least common multiple of 3 and 4.
Examples
72
Check if it’s divisible by 3. The sum of its digits is 7 + 2 = 9, which is divisible by 3. Now, check if it’s divisible by 4, which means the last two digits, 72, must be divisible by 4. Since 72 is divisible by 4, we can conclude that 72 is divisible by 12.
300
Check if it’s divisible by 3. The sum of its digits is 3 + 0 + 0 = 3, which is divisible by 3. Now, check if it’s divisible by 4, which means the last two digits, 00, are divisible by 4 (any number ending in 00 is divisible by 4). Therefore, 300 is divisible by 12.
95
The sum of its digits is 9 + 5 = 14, which is not divisible by 3. Therefore, 95 is not divisible by 12.
216
The sum of its digits is 2 + 1 + 6 = 9, which is divisible by 3. The last two digits, 16, are divisible by 4. Since 16 is divisible by 4, we can conclude that 216 is divisible by 12.
Factors can be useful
Factors can be useful in mathematics, since they are numbers or algebraic expressions that divide another number or expression equally, that is, without any remainder. To calculate the factors of a number, we must first determine which numbers can be divided into it. This is where the divisibility rules come in handy!
For example,
Divisibility for prime numbers
Prime numbers less than 20 and greater than 10 are divided using intermediate divisibility criteria. The divisibility of the prime numbers 2, 3, 5, 7 and 11 has already been tested.
Rules of 13
When we divide a number by 13 and the remainder of the number is 0, it is divisible by 13. Without doing a long division, the 13 divisibility test allows us to quickly determine if a number is divisible by 13. All first, we need to multiply the unit digit by 4 according to the divisibility rule of 13. Then, excluding the digit at the unit location, we add the product to the rest of the number to its left. If the result is a number divisible by 13, the original whole number is also divisible by 13.
Rules of 17
When 17 completely divides a whole number, it is said to be divisible by 17. First, we need to multiply the unit digit by 5 according to the divisibility rule of 17. Then, excluding the unit digit location of the unit, we subtract the product from the rest of the number to its left. If the difference gives a number divisible by 17, the original number is also divisible by 17.
Rules of 19
When we divide a number by 19 and get 0 as the remainder, we say that the number is divisible by 19. First, we need to multiply the ones digit by 2 according to the divisibility rule of 19. Then, excluding the number at the unit location, we add the product to the rest of the number to its left. If the result is a number divisible by 19, the original number is also divisible by 19.
Fun activities with divisibility rules
Divisibility rules are a fundamental mathematical concept that can be used to simplify and solve complex problems. While divisibility rules can be taught and learned in a traditional classroom setting, there are many fun and engaging activities that can be used to make the learning process more enjoyable and effective.
Divisibility Rules Game
- Materials: Whiteboard or projector, markers or pens, paper, and small prizes (optional)
- Instructions:
- Write down different divisibility rules on the whiteboard or projector.
- Divide the class into small teams.
- Give each team a piece of paper and a pen.
- Call out a number and have the teams race to see which team can correctly identify the first divisibility rule that applies to the number.
- The first team to correctly identify the rule wins the round.
- Continue playing until all of the numbers have been called out.
- The team with the most points at the end of the game wins.
Divisibility Rules Scavenger Hunt
- Materials: Paper, pens, and different divisibility rules written on slips of paper
- Instructions:
- Hide the divisibility rules slips of paper around the classroom or school.
- Give each student a piece of paper and a pen.
- Have the students search for the divisibility rules slips of paper and write them down on their pieces of paper.
- Once all of the divisibility rules have been found, have the students sit down and try to solve as many divisibility problems as they can in a set amount of time.
- The student with the most correct answers at the end of the time limit wins.
Divisibility Rules Poster
- Materials: Poster board, markers or pens, and other creative materials (optional)
- Instructions:
- Design a poster that includes all of the different divisibility rules.
- Use creative materials, such as drawings, stickers, and glitter, to decorate your poster.
- Hang your poster in a prominent place in the classroom or school so that students can easily refer to it.
Divisibility Rules Memory Game
- Materials: Paper, pens, and scissors
- Instructions:
- Write down different divisibility rules on one side of slips of paper and examples of numbers that follow the rule on the other side.
- Cut the slips of paper in half so that you have two matching cards for each divisibility rule.
- Mix up the cards and place them face down on a table.
- Have the students take turns flipping over two cards at a time.
- If the cards match, the student keeps them.
- If the cards don’t match, the student flips them back over and the next student takes a turn.
- The student with the most cards at the end of the game wins.
These are just a few professionalized versions of divisibility rules activities. By using your creativity and knowledge of your students, you can develop many other fun and engaging activities that will help your students learn and understand divisibility rules.
- Make sure that the activities are aligned with your curriculum and learning objectives.
- Provide students with clear instructions and expectations.
- Use a variety of activities to keep students engaged and motivated.
- Differentiate the activities to meet the needs of all learners.
- Assess student learning to ensure that they are meeting the learning objectives.
By following these tips, you can create divisibility rules activities that are both fun and educational.
Divisibility rules are a fun and easy way to learn math. By following the tips above, you can learn divisibility rules quickly and accurately. So start practicing today and see how much fun you can have while learning math!
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