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Let's solve this step-by-step.

### Part (a): Prime Factor Decomposition of 252

We need to find the prime factorization of 252.

1. Start by dividing by the smallest prime, 2:
   \[
   252 \div 2 = 126
   \]
2. Divide 126 by 2 again:
   \[
   126 \div 2 = 63
   \]
3. Now, 63 is not divisible by 2, so try the next smallest prime, 3:
   \[
   63 \div 3 = 21
   \]
4. Divide 21 by 3:
   \[
   21 \div 3 = 7
   \]
5. 7 is a prime number.

So, the prime factorization of 252 is:
\[
252 = 2^2 \times 3^2 \times 7
\]

### Part (b): Finding Factors of 252

Now, based on the prime factorization of 252, we need to determine which two numbers from the list are factors of 252. 

- The prime factorization of 252 is \( 2^2 \times 3^2 \times 7 \).
- To be a factor of 252, a number's prime factorization must be a subset of the prime factors of 252.

Let's analyze the given numbers:

1. **22 = \( 2 \times 11 \)**: This is not a factor of 252 because it contains 11, which is not a factor of 252.
2. **14 = \( 2 \times 7 \)**: This is a factor of 252, as both 2 and 7 are present in the prime factorization of 252.
3. **24 = \( 2^3 \times 3 \)**: This is not a factor of 252 because it has \( 2^3 \), while 252 only has \( 2^2 \).
4. **9 = \( 3^2 \)**: This is a factor of 252, as \( 3^2 \) matches the prime factorization of 252.
5. **49 = \( 7^2 \)**: This is not a factor of 252 because 49 contains \( 7^2 \), while 252 only has a single 7.

### Conclusion:
The two numbers that are factors of 252 are **14** and **9**.

---

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. What is the greatest common divisor (GCD) of 252 and 49?
2. How would you find the least common multiple (LCM) of 14 and 252?
3. Can you express the factors of 252 in terms of their greatest common divisor?
4. How do you determine if a larger number is a factor of another without prime factorization?
5. What would be the prime factorization of the product of 14 and 9?

Tip: To check if a number is a factor, verify that all its prime factors are contained in the prime factorization of the larger number!

a) Work out the prime factor decomposition of 252. Give your answer in index form. b) The prime factor decompositions of five numbers are shown below. Use your answer to part a) to work out which two of these numbers are factors of 252.

Learn how to decompose 252 into its prime factors in index form and determine which two given numbers are factors of 252. This problem covers prime factorization, factor identification, and the fundamental theorem of arithmetic. Suitable for Grades 6-8.

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