Question

What is the greatest common factor of 77, 63, 7, and 98?

Answer 1

To find the greatest common factor (GCF) of the numbers 77, 63, 7, and 98, we can first determine the prime factorizations of each number:

1. **77**: The primes factors of 77 are:
\[
77 = 7 \times 11
\]

2. **63**: The prime factors of 63 are:
\[
63 = 3^2 \times 7
\]

3. **7**: The prime factorization of 7 is:
\[
7 = 7^1
\]

4. **98**: The prime factors of 98 are:
\[
98 = 7 \times 14 = 7 \times 2 \times 7 = 7^2 \times 2
\]

Next, we identify the prime factors common to all four numbers. The only prime factor that appears in all of them is \(7\).

Now, we determine the lowest power of this prime factor \(7\) across all these factorizations:
- In 77, the power of \(7\) is \(1\) (from \(7^1\)).
- In 63, the power of \(7\) is \(1\).
- In 7, the power of \(7\) is \(1\).
- In 98, the power of \(7\) is \(2\).

The lowest power of \(7\) that appears in all four numbers is \(7^1\).

Finally, the greatest common factor of 77, 63, 7, and 98 is:
\[
\text{GCF} = 7^1 = 7
\]

Thus, the greatest common factor of 77, 63, 7, and 98 is \(\boxed{7}\).