Find the LCM of 72, 78, and 90.
4680
To find the Least Common Multiple (LCM) of 72, 78, and 90, we can use the method of prime factorization.
- First, let's find the prime factorization of each number:
- 72: \(72 = 2^3 \times 3^2\)
- 78: \(78 = 2^1 \times 3^1 \times 13^1\)
- 90: \(90 = 2^1 \times 3^2 \times 5^1\)
- Identify the highest power of each prime number present in the factorizations:
- \(2\): Maximum power is \(2^3\) (from 72).
- \(3\): Maximum power is \(3^2\) (from 72 and 90).
- \(5\): Maximum power is \(5^1\) (from 90).
- \(13\): Maximum power is \(13^1\) (from 78).
- Multiply these highest powers to get the LCM: \(LCM = 2^3 \times 3^2 \times 5^1 \times 13^1\).
- Calculate the product:
- \(2^3 = 8\)
- \(3^2 = 9\)
- \(5^1 = 5\)
- \(13^1 = 13\)
Now, we simplify the calculations step by step:
- \(8 \times 9 = 72\)
- \(72 \times 5 = 360\)
- \(360 \times 13 = 4680\)
Therefore, the LCM of 72, 78, and 90 is 4680.
Hence, the correct answer is 4680.
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