Final answer:
The prime factors of 75 are 3 and 5², and those of 135 are 3³ and 5. The highest common factor (HCF) is 15, and the least common multiple (LCM) is 405.
Step-by-step explanation:
First, let's find the prime factors of 75 and 135. For 75, we can do successive divisions by prime numbers:
- 75 ÷ 3 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So, the prime factors of 75 are 3 and 5, with 5 being repeated twice (i.e., 5²).
Now, let's factor 135:
- 135 ÷ 3 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
So, the prime factors of 135 are 3, repeated three times (i.e., 3³), and 5.
The highest common factor (HCF) is the product of the smallest powers of common primes, which for these numbers is 3¹ × 5¹ = 15.
The least common multiple (LCM) is the product of the highest powers of all primes present: 3³ × 5² = 405.
Therefore, the prime factors of 75 are 3 and 5², the prime factors of 135 are 3³ and 5, the LCM of 75 and 135 is 405, and the HCF is 15.