To simplify the fraction \( \frac{665}{99976} \), we can check if there is a common factor between the numerator and the denominator.
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Finding the GCD (Greatest Common Divisor):
We'll use the Euclidean algorithm to find the GCD of 665 and 99976.
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First, divide 99976 by 665: \[ 99976 \div 665 \approx 150.3 \quad \text{(take the integer part, which is 150)} \] \[ 99976 - 150 \times 665 = 99976 - 99750 = 226 \]
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Now, apply the algorithm again with 665 and 226: \[ 665 \div 226 \approx 2.9 \quad \text{(take the integer part, which is 2)} \] \[ 665 - 2 \times 226 = 665 - 452 = 213 \]
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Now apply the algorithm with 226 and 213: \[ 226 \div 213 \approx 1.06 \quad \text{(take the integer part, which is 1)} \] \[ 226 - 1 \times 213 = 226 - 213 = 13 \]
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Finally, apply the algorithm with 213 and 13: \[ 213 \div 13 = 16.3846 \quad \text{(take the integer part, which is 16)} \] \[ 213 - 16 \times 13 = 213 - 208 = 5 \]
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Now, apply the algorithm with 13 and 5: \[ 13 \div 5 = 2.6 \quad \text{(take the integer part, which is 2)} \] \[ 13 - 2 \times 5 = 13 - 10 = 3 \]
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Next, apply the algorithm with 5 and 3: \[ 5 \div 3 = 1.66667 \quad \text{(take the integer part, which is 1)} \] \[ 5 - 1 \times 3 = 5 - 3 = 2 \]
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Finally, apply the algorithm with 3 and 2: \[ 3 \div 2 = 1.5 \quad \text{(take the integer part, which is 1)} \] \[ 3 - 1 \times 2 = 3 - 2 = 1 \]
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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.
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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 \( \frac{665}{99976} \).