Question
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Answer
The LCM of 15 and 20 is 60.
Solution
1. Express each number as a product of prime factors:
- \(15 = 3 \times 5\)
- \(20 = 2^2 \times 5\)
2. Identify all unique prime factors and choose the highest power of each:
- For \(2\): highest power is \(2^2\) (from \(20\)).
- For \(3\): highest power is \(3\) (from \(15\)).
- For \(5\): highest power is \(5\) (common in both, but coefficient is \(5\)).
3. Multiply these together to find the LCM:
- \(\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5\)
- \(\text{LCM} = 60\)
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Simplify this solution Beyond the Answer
To find the least common multiple (LCM) of 15 and 20, we can start by using the prime factorization method. The number 15 factors into 3 and 5 (3 x 5), while 20 factors into 2, 2, and 5 (2 x 2 x 5). The LCM is found by taking the highest power of each prime factor from the numbers. So, from 15 we take 3^1 and 5^1, and from 20 we take 2^2. Therefore, the LCM is 2^2 x 3^1 x 5^1 = 4 x 3 x 5 = 60. So, the LCM of 15 and 20 is 60! Also, if you remember, the LCM is also the smallest number that is a multiple of both original numbers. So, counting up, we see that 60 is indeed a multiple of both 15 (4 times) and 20 (3 times), confirming our calculation!
