The LCM of 30 and 45 is:

(A) 15

(B) 30

(C) 45

(D) 90

To find the least common multiple (LCM) of 30 and 45, we can use a couple of different methods, but let's focus on the prime factorization approach, as it clearly illustrates how we arrive at the answer.

Step 1: Prime Factorization

First, we need to break down both numbers into their prime factors:

• 30: The prime factors of 30 are 2, 3, and 5. So, we can express it as 2 × 3 × 5.
• 45: The prime factors of 45 are 3 and 5. We can express it as 3² × 5 (since 45 = 3 × 3 × 5).

Step 2: Identify the Highest Powers

Next, we take each prime factor that appears in either number and use the highest power of that factor:

• For the prime factor 2, the highest power is 2¹ (from 30).
• For the prime factor 3, the highest power is 3² (from 45).
• For the prime factor 5, the highest power is 5¹ (from both numbers).

Step 3: Calculate the LCM

Now we multiply these highest powers together to find the LCM:

LCM = 2¹ × 3² × 5¹ = 2 × 9 × 5

Calculating this step-by-step:

• 2 × 9 = 18
• 18 × 5 = 90

Final Result

Therefore, the least common multiple of 30 and 45 is 90. So, the correct answer is (D) 90.

This method not only gives us the answer but also helps us understand the relationship between the numbers through their prime factors. If you have any further questions about LCM or related concepts, feel free to ask!