When you want to simplify fractions, it’s super important to know how to find the greatest common divisor (GCD) of two numbers.

The GCD is the biggest number that can divide both numbers without leaving anything left over.

Here’s a simple guide that Year 7 students can follow to find the GCD.

Step 1: Start with Two Numbers

First, pick the two numbers you want to check.

Let’s use $24$ and $36$ as our example.

Step 2: List the Factors

Next, find all the factors of each number.

A factor is a whole number that can divide another number evenly.

• Factors of 24:
  • $1 \times 24$
  • $2 \times 12$
  • $3 \times 8$
  • $4 \times 6$

So, the factors of $24$ are: $1, 2, 3, 4, 6, 8, 12, 24$.

• Factors of 36:
  • $1 \times 36$
  • $2 \times 18$
  • $3 \times 12$
  • $4 \times 9$
  • $6 \times 6$

So, the factors of $36$ are: $1, 2, 3, 4, 6, 9, 12, 18, 36$.

Step 3: Find the Common Factors

Now, look at the lists of factors and find the numbers that appear in both lists.

For our numbers:

• Common factors of $24$ and $36$: $1, 2, 3, 4, 6, 12$.

Step 4: Pick the Greatest Common Factor

From the common factors, choose the largest one.

For $24$ and $36$, the greatest common factor (or GCD) is $12$.

Step 5: Check Your Answer

It’s always a good idea to double-check your answer.

You can do this by dividing both numbers by the GCD and seeing if you get any leftovers.

• For $24$: $24 \div 12 = 2$ (no leftovers)

• For $36$: $36 \div 12 = 3$ (no leftovers)

Since both divisions have no leftovers, our GCD is correct!

Another Method: Prime Factorization

You can also find the GCD using prime factorization.

1. Break each number into prime factors.

   • $24 = 2^3 \times 3^1$
   • $36 = 2^2 \times 3^2$

2. Find the smallest powers of the common prime factors.

   • For the factor $2$: min(3, 2) = $2$.
   • For the factor $3$: min(1, 2) = $1$.

3. Multiply the chosen factors:

   • $GCD = 2^2 \times 3^1 = 4 \times 3 = 12$.

Conclusion

Finding the GCD is a handy skill for simplifying fractions and understanding how two numbers are related.

Whether you list the factors or use prime factorization, with practice, you’ll be able to find the GCD quickly and easily!

Happy calculating!