What is the GCF( Greatest common factor) of 12 36 72
The Correct Answer and Explanation is:
To find the Greatest Common Factor (GCF) of 12, 36, and 72, you first need to find the prime factorizations of these numbers:
Step 1: Prime Factorization
- 12:
- Divide 12 by 2: 12÷2=612 ÷ 2 = 612÷2=6
- Divide 6 by 2: 6÷2=36 ÷ 2 = 36÷2=3
- 3 is prime, so the prime factorization of 12 is:
12=2×2×312 = 2 × 2 × 312=2×2×3 or 22×32^2 × 322×3
- 36:
- Divide 36 by 2: 36÷2=1836 ÷ 2 = 1836÷2=18
- Divide 18 by 2: 18÷2=918 ÷ 2 = 918÷2=9
- Divide 9 by 3: 9÷3=39 ÷ 3 = 39÷3=3
- 3 is prime, so the prime factorization of 36 is:
36=2×2×3×336 = 2 × 2 × 3 × 336=2×2×3×3 or 22×322^2 × 3^222×32
- 72:
- Divide 72 by 2: 72÷2=3672 ÷ 2 = 3672÷2=36
- Divide 36 by 2: 36÷2=1836 ÷ 2 = 1836÷2=18
- Divide 18 by 2: 18÷2=918 ÷ 2 = 918÷2=9
- Divide 9 by 3: 9÷3=39 ÷ 3 = 39÷3=3
- 3 is prime, so the prime factorization of 72 is:
72=2×2×2×3×372 = 2 × 2 × 2 × 3 × 372=2×2×2×3×3 or 23×322^3 × 3^223×32
Step 2: Identify Common Factors
Now, list the common prime factors for 12, 36, and 72:
- For 2, the lowest exponent among the numbers is 222^222 (from 12 and 36).
- For 3, the lowest exponent among the numbers is 313^131 (from 12).
Step 3: Multiply Common Factors
To find the GCF, multiply the common factors with the lowest exponents:GCF=22×3=4×3=12GCF = 2^2 × 3 = 4 × 3 = 12GCF=22×3=4×3=12
Conclusion:
The Greatest Common Factor of 12, 36, and 72 is 12.
This method of finding the GCF is based on the fact that the GCF is the product of the lowest powers of all the prime factors common to the numbers. By identifying the prime factors and comparing their exponents, you can easily determine the greatest factor they all share.
