Factors Of 24 And 36: Common Factors And Greatest Common Factor

Finding the factors of two numbers is a foundational skill in mathematics, especially when you need to simplify fractions or find common denominators. When you look at the numbers 24 and 36, understanding their divisors lets you quickly identify their shared relationships.

To find the factors of 24 and 36, you list every whole number that divides evenly into each value without leaving a remainder. For 24, those numbers are 1, 2, 3, 4, 6, 8, 12, and 24. For 36, the factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Comparing these two sets reveals shared divisors like 1, 2, 3, 4, 6, and 12, with 12 serving as the greatest common factor.

What Are Factors and Multiples?

Students often confuse factors with multiples, but they are entirely different concepts. A factor is a number that divides evenly into another number. A multiple is the product you get when you multiply that number by an integer.

For instance, the number 6 is a factor of 24 because 24 divided by 6 equals 4 with no remainder. Conversely, 24 is a multiple of 6 because 6 multiplied by 4 equals 24.

When working with factors, you are always dealing with numbers that are smaller than or equal to the original number. When working with multiples, you are dealing with numbers that are equal to or larger than the original number. Every positive integer has a finite number of factors, but every integer has an infinite number of multiples.

Step-by-Step Method to Find the Factors of 24

Let's break down the exact process for finding every factor of 24. You don't have to guess or check every number up to 24. You can use systematic division.

Start with the number 1 and work your way up.

1. Test 1: 24 divided by 1 is 24. This gives us our first pair: 1 and 24.
2. Test 2: 24 divided by 2 is 12. This gives us the pair: 2 and 12.
3. Test 3: 24 divided by 3 is 8. This gives us the pair: 3 and 8.
4. Test 4: 24 divided by 4 is 6. This gives us the pair: 4 and 6.
5. Test 5: 24 divided by 5 does not yield a whole number. Skip it.
6. Test 6: 24 divided by 6 is 4. We have already reached this pair, which means we have found all the unique factors.

Listing these in numerical order gives us: 1, 2, 3, 4, 6, 8, 12, and 24.

Step-by-Step Method to Find the Factors of 36

Apply the exact same systematic approach to find the factors of 36. Testing integers starting from 1 keeps you organized.

1. Test 1: 36 divided by 1 is 36. Pair: 1 and 36.
2. Test 2: 36 divided by 2 is 18. Pair: 2 and 18.
3. Test 3: 36 divided by 3 is 12. Pair: 3 and 12.
4. Test 4: 36 divided by 4 is 9. Pair: 4 and 9.
5. Test 5: 36 divided by 5 does not work. Skip it.
6. Test 6: 36 divided by 6 is 6. This is a square root factor, meaning 6 multiplied by itself equals 36.

Listing these numbers in numerical order gives us: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Notice how 36 has an odd number of total factors because 6 is paired with itself. This is a common trait for square numbers. You can read more about integer properties on MathWorld.

Finding Common Factors

Once you have both lists, comparing them is straightforward. The common factors are the numbers that appear on both lists.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Comparing these two sets shows that the shared numbers are 1, 2, 3, 4, 6, and 12. These are the common factors of 24 and 36. Any fraction with 24 in the numerator and 36 in the denominator can be simplified using any of these shared divisors, though using the largest one simplifies the fraction in a single step.

Finding the Greatest Common Factor (GCF)

The Greatest Common Factor is simply the largest value found in your list of common factors. For 24 and 36, that number is 12.

Finding the GCF is useful when simplifying fractions. If you have the fraction 24 over 36, dividing both the numerator and the denominator by their GCF (12) immediately reduces the fraction to 2/3.

Prime Factorization Method

While listing all factors works well for smaller numbers, prime factorization is much faster for larger numbers. Let's find the GCF of 24 and 36 using prime trees.

For 24:

• 24 broken down gives 2 times 12.
• 12 breaks down into 2 times 6.
• 6 breaks down into 2 times 3.
• The prime factorization of 24 is 2 times 2 times 2 times 3, or two cubed times three.

For 36:

• 36 broken down gives 2 times 18.
• 18 breaks down into 2 times 9.
• 9 breaks down into 3 times 3.
• The prime factorization of 36 is 2 times 2 times 3 times 3, or two squared times three squared.

To find the GCF using prime factorization, you take the lowest power of every common prime factor. Both numbers share prime factors of 2 and 3.

• For 2, the lowest power present in both breakdowns is two squared (4).
• For 3, the lowest power present in both breakdowns is three to the first power (3).

Multiply these together: 4 times 3 equals 12. The GCF is 12. For deeper math guides and curricula, check resources provided by the National Council of Teachers of Mathematics.

Alternative Methods: Euclidean Algorithm

When numbers get very large, listing factors or building prime trees becomes tedious. Mathematicians use the Euclidean Algorithm instead. This method relies on repeated division and remainders.

Take our numbers, 36 and 24.

1. Divide 36 by 24. You get 1 with a remainder of 12.
2. Take your previous divisor (24) and divide it by the remainder (12). You get 2 with a remainder of 0.
3. Once your remainder is 0, the last non-zero remainder is your GCF.

The last non-zero remainder here is 12. This confirms that 12 is indeed the greatest common factor. This method scales effectively for numbers in the thousands.

Common Mistakes to Avoid

People often make a few predictable errors when finding factors. Keep these in mind so you don't lose points on homework or make calculation errors.

• Forgetting the number 1 and the number itself. Every integer has at least two factors: 1 and itself. Don't skip them.
• Confusing factors with multiples. If you write down 48 as a factor of 24, you're listing a multiple. Factors are always smaller than or equal to the target number (except for negative numbers, which aren't typically used in basic factor lists).
• Missing pairs. Using the division method ensures you catch factor pairs. If you find that 3 is a factor of 24, immediately write down its partner, 8.

Frequently Asked Questions

What is the least common multiple of 24 and 36?

While the GCF is 12, the least common multiple (LCM) is 72. You can find the LCM by multiplying the two original numbers together and dividing by their GCF. So, 24 times 36 equals 864, and 864 divided by 12 is 72.

Are 24 and 36 relatively prime?

No. Two numbers are relatively prime if their only common factor is 1. Since 24 and 36 share common factors of 2, 3, 4, 6, and 12, they are not relatively prime.

Can negative numbers be factors?

Yes, technically -1 and -2 are factors of 24 because multiplying them by -24 or -12 yields a positive 24. However, in standard elementary and middle school math problems, we focus strictly on positive whole numbers.

How many total factors does 36 have?

Counting them up from our list (1, 2, 3, 4, 6, 9, 12, 18, 36), the number 36 has a total of 9 factors. This is unusually high for a number its size because 36 is a highly composite number with many divisors.