72 and 288 Common factor/s: GCF: Common factor/s: 15. 51 and 183 GCF: 16. 18, 36, and 72 Common factor/s: GCF: _ 17. 64, 80, and 240 Common factor/s: GCF:
The Correct Answer and Explanation is:
Let’s solve the problems one by one, identifying both the common factors and the greatest common factor (GCF) for each set of numbers.
14. 72 and 288
Common factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
GCF: 72
Explanation:
To find the common factors, we list all factors of each number.
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 288 include all of the above since 72 is a factor of 288.
Since all factors of 72 are also in 288, the common factors are just the factors of 72. The greatest among them is 72.
15. 51 and 183
Common factors: 1, 3
GCF: 3
Explanation:
- Factors of 51: 1, 3, 17, 51
- Factors of 183: 1, 3, 61, 183
Only 1 and 3 appear in both lists. The greatest of these is 3.
16. 18, 36, and 72
Common factors: 1, 2, 3, 6, 9, 18
GCF: 18
Explanation:
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Common factors across all three sets are 1, 2, 3, 6, 9, and 18. The largest is 18.
17. 64, 80, and 240
Common factors: 1, 2, 4, 8, 16
GCF: 16
Explanation:
- Factors of 64: 1, 2, 4, 8, 16, 32, 64
- Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
- Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
The common factors are 1, 2, 4, 8, and 16. The greatest is 16.
Conclusion:
Finding common factors involves listing the full factor sets of each number and identifying which values appear in all sets. The greatest value in the intersection of these sets is the greatest common factor (GCF). The GCF is useful in simplifying fractions, finding ratios, and solving problems in number theory.
