Divisibility rules are simple shortcuts that help us quickly determine if one number is divisible by another without needing to perform division. By applying these rules, we can see at a glance if a number can be evenly divided by factors like 2, 3, 5, or 10, among others. This knowledge is invaluable in arithmetic, number theory, and problem-solving, providing efficiency in calculations and helping to break down large numbers into their components.
This article will delve into the divisibility rules for numbers 2 through 11, illustrating each rule with examples to show how these shortcuts work in practice.
Divisibility Rule for 2
A number is divisible by 2 if its last digit is even. An even number ends in 0, 2, 4, 6, or 8, which means it can be divided by 2 without a remainder.
- Example:
– 38 is divisible by 2 because it ends in 8, which is an even number.
– 45 is not divisible by 2 because it ends in 5, which is an odd number.
This rule applies universally to any size of number, making it easy to determine divisibility by 2.
Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3. This rule helps simplify larger numbers, where adding the digits gives a manageable number for division.
- Example:
– 123: The sum of the digits is 
. Since 6 is divisible by 3, 123 is also divisible by 3.
– 87: The sum of the digits is 
. Since 15 is divisible by 3, 87 is also divisible by 3.
– 146: The sum of the digits is 
. Since 11 is not divisible by 3, 146 is not divisible by 3.
Divisibility Rule for 4
A number is divisible by 4 if the last two digits form a number that is divisible by 4. This rule allows you to ignore all but the last two digits of the number.
- Example:
– 312: The last two digits are 12, which is divisible by 4, so 312 is also divisible by 4.
– 1048: The last two digits are 48, which is divisible by 4, so 1048 is divisible by 4.
– 273: The last two digits are 73, which is not divisible by 4, so 273 is not divisible by 4.
Divisibility Rule for 5
A number is divisible by 5 if its last digit is 0 or 5. This is one of the simplest divisibility rules and can be applied quickly to any number.
- Example:
– 135 ends in 5, so it is divisible by 5.
– 490 ends in 0, so it is also divisible by 5.
– 267 ends in 7, so it is not divisible by 5.
Divisibility Rule for 6
A number is divisible by 6 if it meets the divisibility rules for both 2 and 3. In other words, the number must be even (divisible by 2) and the sum of its digits must be divisible by 3.
- Example:
– 72: It is even (ends in 2), and the sum of its digits is 
, which is divisible by 3. Therefore, 72 is divisible by 6.
– 114: It is even, and the sum of its digits is 
, which is divisible by 3. So, 114 is divisible by 6.
– 245: It is not even (ends in 5), so it is not divisible by 6, even though the sum of its digits (2 + 4 + 5 = 11) is not divisible by 3 either.
Divisibility Rule for 7
The rule for 7 involves a slightly more complex process:
1. Double the last digit of the number.
2. Subtract this doubled value from the rest of the number.
3. If the result is divisible by 7, then the original number is also divisible by 7.
- Example:
– 203: Double the last digit (3 × 2 = 6) and subtract it from the remaining number (20 – 6 = 14). Since 14 is divisible by 7, 203 is also divisible by 7.
– 301: Double the last digit (1 × 2 = 2) and subtract it from the remaining number (30 – 2 = 28). Since 28 is divisible by 7, 301 is also divisible by 7.
– 342: Double the last digit (2 × 2 = 4) and subtract it from the remaining number (34 – 4 = 30). Since 30 is not divisible by 7, 342 is not divisible by 7.
This rule can be repeated if necessary for large numbers.
Divisibility Rule for 8
A number is divisible by 8 if the last three digits form a number that is divisible by 8. This rule works well for large numbers where focusing on the last three digits alone can simplify the calculation.
- Example:
– 7128: The last three digits are 128, which is divisible by 8, so 7128 is divisible by 8.
– 15024: The last three digits are 024 (or simply 24), which is not divisible by 8, so 15024 is not divisible by 8.
– 4096: The last three digits are 096, which is divisible by 8, so 4096 is divisible by 8.
Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9. This rule is similar to the divisibility rule for 3 but uses 9 as the divisor.
- Example:
– 729: The sum of the digits is 
, which is divisible by 9, so 729 is divisible by 9.
– 153: The sum of the digits is 
, which is divisible by 9, so 153 is divisible by 9.
– 214: The sum of the digits is 
, which is not divisible by 9, so 214 is not divisible by 9.
Divisibility Rule for 10
A number is divisible by 10 if its last digit is 0. This rule is particularly simple and applies universally to all whole numbers.
- Example:
– 450 ends in 0, so it is divisible by 10.
– 1230 also ends in 0, so it is divisible by 10.
– 567 does not end in 0, so it is not divisible by 10.
Divisibility Rule for 11
A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is a multiple of 11 (including 0). This rule might sound complex, but it’s easy to apply with a little practice.
1. Add the digits in the odd positions.
2. Add the digits in the even positions.
3. Subtract the two sums.
4. If the result is 0 or a multiple of 11, the number is divisible by 11.
- Example:
– 121: The sum of the odd-position digits is 
, and the sum of the even-position digit is 
. The difference is 
, which is divisible by 11, so 121 is divisible by 11.
– 253: The sum of the odd-position digits is 
, and the even-position digit is 
. The difference is 
, so 253 is divisible by 11.
– 1441: The sum of the odd-position digits is 
, and the sum of the even-position digits is 
. The difference is , which means 1441 is divisible by 11.
Importance of Divisibility Rules
Divisibility rules play a fundamental role in arithmetic and number theory by simplifying complex problems, particularly when factoring large numbers, simplifying fractions, and solving divisibility questions. Additionally, divisibility rules are valuable in fields such as cryptography, where prime factorization and modular arithmetic are essential.
In addition to their importance in mathematics, divisibility rules are practical tools for quick calculations, estimation, and error-checking in everyday life. For example, when dividing up items among groups, divisibility rules help determine if an even distribution is possible without performing full division.
Conclusion: Mastering Divisibility Rules
Divisibility rules are invaluable tools for anyone working with numbers, from students to professionals in technical fields. By understanding and applying these rules, we can quickly assess divisibility, simplify mathematical operations, and identify factors of large numbers without tedious calculations. Whether determining if a number is even, checking for factors, or solving more complex divisibility puzzles, mastering these rules is a powerful way to enhance mathematical fluency and efficiency.
