The GCF of 26 39 and 65
The Correct Answer and Explanation is:
To determine the Greatest Common Factor (GCF) of the numbers 26, 39, and 65, we begin by finding the prime factorizations of each number:
- 26:
26 is divisible by 2 (since it is an even number), so: 26=2×1326 = 2 \times 1326=2×13 Therefore, the prime factorization of 26 is 2×132 \times 132×13. - 39:
39 is divisible by 3 (since the sum of the digits, 3 + 9 = 12, is divisible by 3), so: 39=3×1339 = 3 \times 1339=3×13 Therefore, the prime factorization of 39 is 3×133 \times 133×13. - 65:
65 is divisible by 5 (since it ends in a 5), so: 65=5×1365 = 5 \times 1365=5×13 Therefore, the prime factorization of 65 is 5×135 \times 135×13.
Step 2: Identify the common factors
Now, we examine the prime factorizations of the three numbers:
- 26: 2×132 \times 132×13
- 39: 3×133 \times 133×13
- 65: 5×135 \times 135×13
The only common factor across all three numbers is 13.
Step 3: Conclusion
Since 13 is the only factor that appears in all three numbers, the Greatest Common Factor (GCF) of 26, 39, and 65 is 13.
Why this works:
The GCF is the largest number that divides each of the given numbers exactly. In this case, 13 is the largest number that divides 26, 39, and 65 without leaving any remainder.
