To find the highest common factor (HCF) of the numbers 12, 30, 42, and 72, we can first find the prime factorization of each number:
- 12: \(2^2 \times 3\)
- 30: \(2 \times 3 \times 5\)
- 42: \(2 \times 3 \times 7\)
- 72: \(2^3 \times 3^2\)
Next, we identify the common prime factors and their lowest powers in each factorization:
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The prime factor \(2\) appears in all numbers:
- In 12: \(2^2\)
- In 30: \(2^1\)
- In 42: \(2^1\)
- In 72: \(2^3\)
The lowest power of \(2\) is \(2^1\).
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The prime factor \(3\) also appears in all numbers:
- In 12: \(3^1\)
- In 30: \(3^1\)
- In 42: \(3^1\)
- In 72: \(3^2\)
The lowest power of \(3\) is \(3^1\).
The prime factors that are common to all the numbers are \(2\) and \(3\).
Multiplying the lowest powers gives us:
\[ HCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]
Therefore, the highest common factor of 12, 30, 42, and 72 is 6.