What is the greatest common factor of 4k, 18k4, and 12?

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To find the greatest common factor (GCF) of the expressions 4k, 18k^4, and 12, we first need to determine the factors of each term.

Step 1: Factor Each Term

  • 4k: The factors are 4 and k, which can be broken down to 2 × 2 and includes the variable k.
  • 18k^4: The factors are 18 and k4, which can be broken down to 2 × 3 × 3 and includes the variable k raised to the 4th power.
  • 12: The factors are 12, which can be broken down to 2 × 2 × 3.

Step 2: Identify Common Factors

Next, we look for the common numerical factors among these terms:

  • The prime factorization of 4k is 2^2 × k.
  • The prime factorization of 18k^4 is 2 × 3^2 × k^4.
  • The prime factorization of 12 is 2^2 × 3.

Step 3: Determine the Lowest Powers of Common Factors

The common prime factors are 2 and 3:

  • For the number 2, the lowest exponent from the factorizations is from 18k^4, which is 21.
  • For the number 3, the lowest exponent is from 12, which is 31.
  • For the variable k, the lowest exponent is from 4k, which is k1.

Step 4: Calculate the GCF

The GCF is then calculated by multiplying the common factors with their lowest powers:

GCF = 21 × 31 × k1 = 2 × 3 × k = 6k

Conclusion

Therefore, the greatest common factor of 4k, 18k^4, and 12 is 6k.

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