Question

665/9,9976 simplified

Answer 1

To simplify the fraction \( \frac{665}{99976} \), we can check if there is a common factor between the numerator and the denominator.

1. **Finding the GCD (Greatest Common Divisor)**:

We'll use the Euclidean algorithm to find the GCD of 665 and 99976.

- First, divide 99976 by 665:
\[
99976 \div 665 \approx 150.3 \quad \text{(take the integer part, which is 150)}
\]
\[
99976 - 150 \times 665 = 99976 - 99750 = 226
\]

- Now, apply the algorithm again with 665 and 226:
\[
665 \div 226 \approx 2.9 \quad \text{(take the integer part, which is 2)}
\]
\[
665 - 2 \times 226 = 665 - 452 = 213
\]

- Now apply the algorithm with 226 and 213:
\[
226 \div 213 \approx 1.06 \quad \text{(take the integer part, which is 1)}
\]
\[
226 - 1 \times 213 = 226 - 213 = 13
\]

- Finally, apply the algorithm with 213 and 13:
\[
213 \div 13 = 16.3846 \quad \text{(take the integer part, which is 16)}
\]
\[
213 - 16 \times 13 = 213 - 208 = 5
\]

- Now, apply the algorithm with 13 and 5:
\[
13 \div 5 = 2.6 \quad \text{(take the integer part, which is 2)}
\]
\[
13 - 2 \times 5 = 13 - 10 = 3
\]

- Next, apply the algorithm with 5 and 3:
\[
5 \div 3 = 1.66667 \quad \text{(take the integer part, which is 1)}
\]
\[
5 - 1 \times 3 = 5 - 3 = 2
\]

- Finally, apply the algorithm with 3 and 2:
\[
3 \div 2 = 1.5 \quad \text{(take the integer part, which is 1)}
\]
\[
3 - 1 \times 2 = 3 - 2 = 1
\]

- And lastly, apply the algorithm with 2 and 1:
\[
2 \div 1 = 2
\]
\[
2 - 2 \times 1 = 2 - 2 = 0
\]

Since the last non-zero remainder is 1, this indicates the GCD of 665 and 99976 is 1.

2. **Conclusion**:

Since the GCD is 1, the fraction \( \frac{665}{99976} \) is already in its simplest form.

Therefore, the simplified form of \( \frac{665}{99976} \) is still \( \frac{665}{99976} \).