Highest common factor of 12 30 42 72

1 answer

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:

  • 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 \(2^1\).

  • 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.