Question

Find the prime factors of 75 and 135. Then, calculate their least common multiple (LCM) and highest common factor (HCF).

a) Prime factors of 75: 3, 5; Prime factors of 135: 3, 5; LCM: 15, HCF: 3
b) Prime factors of 75: 2, 3, 5; Prime factors of 135: 3, 5, 9; LCM: 405, HCF: 15
c) Prime factors of 75: 5, 15; Prime factors of 135: 3, 5, 9; LCM: 225, HCF: 5
d) Prime factors of 75: 2, 3, 5; Prime factors of 135: 3, 5, 3; LCM: 405, HCF: 5

Answer 1

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.