Malaysia Collins What is the least common multiple of 15 and 25? (A) 50 (B) 75 C) 5
(D) 40
The correct answer and explanation is:
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
Let’s find the LCM of 15 and 25.
Step 1: Prime Factorization
- 15 = 3 × 5
- 25 = 5 × 5 (or 5²)
Step 2: Identify the highest powers of each prime factor appearing in either number:
- For prime 3: highest power is 3¹ (from 15)
- For prime 5: highest power is 5² (from 25)
Step 3: Multiply these highest powers together to get the LCM:
LCM = 3¹ × 5² = 3 × 25 = 75
Answer: (B) 75
Explanation:
The least common multiple (LCM) is essential when you want to find a common base to work with two different numbers, especially useful in adding or comparing fractions, scheduling events, or solving problems involving repeating cycles.
- Multiples of 15: 15, 30, 45, 60, 75, 90, …
- Multiples of 25: 25, 50, 75, 100, 125, …
The smallest multiple that appears in both lists is 75.
Another method to find the LCM is using the relationship between the Greatest Common Divisor (GCD) and LCM: LCM(a,b)=a×bGCD(a,b)\text{LCM}(a,b) = \frac{a \times b}{\text{GCD}(a,b)}
- GCD of 15 and 25 is 5 (the largest number that divides both evenly).
- Then:
LCM=15×255=3755=75\text{LCM} = \frac{15 \times 25}{5} = \frac{375}{5} = 75
Thus, 75 is the least common multiple of 15 and 25.
Summary:
- The least common multiple of 15 and 25 is 75.
- This is because 75 is the smallest number divisible by both 15 and 25.
- Answer choice (B) 75 is correct.