Certainly! Let's break down this problem step by step.
### Part (a): Write 600 as the product of prime factors.
1. Prime Factorization of 600:
We need to express 600 as a product of prime numbers raised to their respective powers.
The prime factorization of 600 can be done as follows:
- Divide 600 by the smallest prime number, which is 2:
[tex]\( \frac{600}{2} = 300 \)[/tex]
- Divide 300 by 2:
[tex]\( \frac{300}{2} = 150 \)[/tex]
- Divide 150 by 2:
[tex]\( \frac{150}{2} = 75 \)[/tex]
- 75 is not divisible by 2. Move to the next smallest prime number, which is 3:
[tex]\( \frac{75}{3} = 25 \)[/tex]
- 25 is not divisible by 3. Move to the next smallest prime number, which is 5:
[tex]\( \frac{25}{5} = 5 \)[/tex]
- Divide 5 by 5:
[tex]\( \frac{5}{5} = 1 \)[/tex]
Now we have all our prime factors. The prime factors are 2, 3, and 5.
- 600 divided by 2 three times gives 2^3.
- 75 divided by 3 once gives 3^1.
- 25 divided by 5 twice gives 5^2.
2. Index Form Representation:
By combining all the prime factors, 600 can be written in index form as:
[tex]\[
600 = 2^3 \times 3^1 \times 5^2
\][/tex]
### Part (b): Work out the Highest Common Factor (HCF)
In order to find the Highest Common Factor (HCF) of two numbers, we need those numbers. Since no specific numbers are given in the question, we cannot determine the HCF here.
However, here's a brief explanation of how you would work out the HCF if two numbers were given:
1. Prime Factorize Both Numbers:
Express both numbers as the product of their prime factors.
2. Identify Common Prime Factors:
List the prime factors that appear in both factorizations.
3. Choose the Lowest Power of Each Common Prime Factor:
For each common prime factor, choose the lowest power that appears in the factorizations of both numbers.
4. Multiply These Factors Together:
Multiply these selected factors to get the HCF.
Since no specific numbers are provided for part (b) in this case, we will focus on part (a) which results in the factorization of 600:
[tex]\[
600 = 2^3 \times 3^1 \times 5^2
\][/tex]
