p 2 3 53 2# #= q 2 3 52# #=(a) Find the highest common factor (HCF) of p and q.

1. ### Break Down the Problem

To find the highest common factor (HCF) of the two numbers represented in prime factorization:

• $p = 2^3 \cdot 5^2$
• $q = 2^3 \cdot 5^2$

2. ### Relevant Concepts

The highest common factor (HCF), also known as greatest common divisor (GCD), can be found by taking the lowest power of each prime factor that appears in the prime factorizations of both numbers.

3. ### Analysis and Detail

Identifying the prime factors:

• For $p$:

  • $2^3$
  • $5^2$

• For $q$:

  • $2^3$
  • $5^2$

Now, we take the lowest powers:

• For $2$: $\min(3, 3) = 3$
• For $5$: $\min(2, 2) = 2$

Thus, the HCF can be calculated as: $\text{HCF} = 2^{\min(3,3)} \cdot 5^{\min(2,2)} = 2^3 \cdot 5^2$

4. ### Verify and Summarize

Calculating the HCF: $2^3 = 8 \quad \text{and} \quad 5^2 = 25$ Thus, $\text{HCF} = 8 \cdot 25 = 200$

Final Answer

The highest common factor (HCF) of $p$ and $q$ is $200$.