Divisible: The Key to Sharing and Grouping Fairly
The Foundation: Division and Remainders
At its heart, divisibility is about division. When you divide one number, called the dividend, by another number, called the divisor, you get a result called the quotient. If the division is exact, meaning there is nothing left over, then the dividend is divisible by the divisor.
If there is no remainder, we say: Dividend is divisible by Divisor.
For example, if you have 12 cookies and 3 friends, you can perform the division 12 ÷ 3 = 4. Each friend gets 4 cookies, and there are none left over. Therefore, 12 is divisible by 3.
Now, imagine you have 14 cookies and the same 3 friends. 14 ÷ 3 = 4 with a remainder of 2 (because 3 × 4 = 12, and 14 - 12 = 2). Here, 14 is not divisible by 3 because there is a remainder.
Handy Divisibility Rules
You don't always need to perform long division to check for divisibility. Mathematicians have discovered simple rules, or tests, for many numbers. These rules help you quickly determine if a number is divisible by another.
| Divisor | Rule | Example |
|---|---|---|
| 2 | The number is even (its last digit is 0, 2, 4, 6, or 8). | 158 is divisible by 2 because it ends in 8. |
| 3 | The sum of all digits is divisible by 3. | 123 (1+2+3=6) is divisible by 3 because 6 is divisible by 3. |
| 4 | The last two digits form a number divisible by 4. | 1,732 is divisible by 4 because 32 is divisible by 4. |
| 5 | The number ends in 0 or 5. | 9,405 is divisible by 5 because it ends in 5. |
| 6 | The number is divisible by both 2 and 3. | 138 is even (divisible by 2) and 1+3+8=12 (divisible by 3), so it's divisible by 6. |
| 9 | The sum of all digits is divisible by 9. | 4,581 (4+5+8+1=18) is divisible by 9 because 18 is divisible by 9. |
| 10 | The number ends in 0. | 670 is divisible by 10 because it ends in 0. |
Factors, Multiples, and Prime Numbers
Divisibility is directly linked to the ideas of factors and multiples.
Factors (or divisors) are the numbers you can multiply together to get another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because 1 × 12 = 12, 2 × 6 = 12, and 3 × 4 = 12. We can also say that 12 is divisible by each of its factors.
Multiples are the results you get when you multiply a number by integers. The multiples of 3 are 3, 6, 9, 12, 15, ... and so on. Notice that every multiple of 3 is divisible by 3.
This leads us to a special category of numbers: prime numbers. A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. This means it is only divisible by 1 and itself. Examples include 2, 3, 5, 7, 11, and 13. The number 1 is not considered prime because it has only one factor.
Applying Divisibility in Mathematics
Divisibility isn't just a standalone concept; it's a powerful tool used throughout mathematics.
Simplifying Fractions: To simplify a fraction like $ \frac{18}{24} $, you find a number that divides both the numerator (18) and the denominator (24). Both are divisible by 6, so you can divide both by 6 to get the simplified fraction $ \frac{18 \div 6}{24 \div 6} = \frac{3}{4} $.
Finding the Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both. Divisibility rules help you find it. For example, to find the LCM of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 6: 6, 12, 18, 24, ...
The smallest number that appears in both lists is 12. So, the LCM of 4 and 6 is 12.
Finding the Greatest Common Factor (GCF): The GCF is the largest number that divides two numbers evenly. For 18 and 24, the common factors are 1, 2, 3, and 6. The greatest of these is 6, so the GCF is 6.
Divisibility in Everyday Situations
We use divisibility constantly in daily life without even realizing it.
Event Planning: If you are organizing 36 participants into teams of 4, you can quickly see that 36 is divisible by 4 (since 36 ÷ 4 = 9), so you can form 9 equal teams. If someone suggested teams of 5, you would know it's impossible without leaving people out because 36 is not divisible by 5.
Packaging and Distribution: A factory produces 250 cans of soup. They need to pack them into boxes that hold 6 cans each. Is 250 divisible by 6? Using the rule for 6, we check: it's even (so divisible by 2), but the sum of its digits is 2+5+0=7, which is not divisible by 3. Therefore, 250 is not divisible by 6, and some boxes will be incomplete.
Common Mistakes and Important Questions
Q: Is the number 1 divisible by itself?
Yes. When you divide 1 by 1, the quotient is 1 and the remainder is 0. So, 1 is divisible by 1.
Q: Can zero (0) be divisible by any number?
Yes, but with an important caveat. 0 divided by any non-zero number is 0. For example, 0 ÷ 5 = 0, which is an exact division with no remainder. Therefore, 0 is divisible by any number except zero itself (since division by zero is undefined).
Q: A common mistake is to think that a larger number cannot be divisible by a smaller number. Is this true?
No, this is false. Divisibility has nothing to do with which number is larger. For example, 10 is divisible by 5, and 5 is smaller than 10. However, a smaller number can also be divisible by a larger number if we are working with fractions, but in the context of whole numbers, if number A is divisible by number B, then A must be greater than or equal to B (unless A is zero).
The concept of divisibility is a simple yet powerful idea that forms the bedrock of many areas in mathematics. From using quick mental checks with divisibility rules to simplifying fractions and finding common factors, understanding when one number can be evenly divided by another is an essential skill. It connects directly to the fundamental operations of arithmetic and paves the way for learning about prime numbers, least common multiples, and greatest common factors. By mastering divisibility, you gain a valuable tool for both academic success and solving everyday problems efficiently.
Footnote
[1] LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more numbers.