To find the Least Common Multiple (LCM) of two numbers, we can use the prime factorization method or the formula that relates the LCM to the Greatest Common Divisor (GCD):
\[ \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} \]
LCM of 12 and 20
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Prime Factorization:
- 12: \(2^2 \times 3^1\)
- 20: \(2^2 \times 5^1\)
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Taking the highest power of each prime:
- For \(2\): maximum power is \(2^2\)
- For \(3\): maximum power is \(3^1\)
- For \(5\): maximum power is \(5^1\)
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LCM Calculation: \[ \text{LCM}(12, 20) = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]
Therefore, the LCM of 12 and 20 is 60.
LCM of 18 and 30
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Prime Factorization:
- 18: \(2^1 \times 3^2\)
- 30: \(2^1 \times 3^1 \times 5^1\)
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Taking the highest power of each prime:
- For \(2\): maximum power is \(2^1\)
- For \(3\): maximum power is \(3^2\)
- For \(5\): maximum power is \(5^1\)
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LCM Calculation: \[ \text{LCM}(18, 30) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 \]
Therefore, the LCM of 18 and 30 is 90.
In summary:
- The LCM of 12 and 20 is 60.
- The LCM of 18 and 30 is 90.