To simplify the fraction \( \frac{65}{87} \), we need to find the greatest common divisor (GCD) of the numerator (65) and the denominator (87).
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First, we find the prime factors of each number:
- The prime factorization of \( 65 \) is \( 5 \times 13 \).
- The prime factorization of \( 87 \) is \( 3 \times 29 \).
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Since there are no common factors between the prime factorizations of \( 65 \) and \( 87 \), the GCD is 1.
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Since the GCD is 1, \( \frac{65}{87} \) is already in its simplest form.
Thus, the simplified fraction is \( \frac{65}{87} \).