What is the greatest common factor of 42, 126, and 210 ?
The correct answer and explanation is:
To find the greatest common factor (GCF) of 42, 126, and 210, we begin by determining the prime factorization of each number.
- Prime factorization of 42:
- Start with the smallest prime number, 2.
- 42 ÷ 2 = 21.
- Now factor 21: 21 ÷ 3 = 7.
- Since 7 is prime, the prime factorization of 42 is: 2×3×72 \times 3 \times 7.
- Prime factorization of 126:
- 126 ÷ 2 = 63.
- Factor 63: 63 ÷ 3 = 21.
- Factor 21: 21 ÷ 3 = 7.
- So, the prime factorization of 126 is: 2×32×72 \times 3^2 \times 7.
- Prime factorization of 210:
- 210 ÷ 2 = 105.
- Factor 105: 105 ÷ 3 = 35.
- Factor 35: 35 ÷ 5 = 7.
- So, the prime factorization of 210 is: 2×3×5×72 \times 3 \times 5 \times 7.
Now that we have the prime factorizations, we find the common factors between all three numbers. The prime factors they share are:
- 2
- 3
- 7
The greatest common factor is the product of these common prime factors:
2×3×7=422 \times 3 \times 7 = 42.
Therefore, the greatest common factor (GCF) of 42, 126, and 210 is 42.
To verify, we divide each of the original numbers by 42:
- 42 ÷ 42 = 1
- 126 ÷ 42 = 3
- 210 ÷ 42 = 5
Since all results are integers, the GCF is confirmed to be 42.