To simplify the fraction \( \frac{15}{9,976} \), we can first check if both the numerator and the denominator have any common factors.
- The prime factorization of the numerator \( 15 \) is \( 3 \times 5 \).
- For the denominator \( 9,976 \), let's factor it:
- \( 9,976 \) is an even number, so it is divisible by \( 2 \): \[ 9,976 \div 2 = 4,988 \] \[ 4,988 \div 2 = 2,494 \] \[ 2,494 \div 2 = 1,247 \]
- Now, \( 1,247 \) needs to be factored further. It is not even, so check for divisibility by \( 3 \): \[ 1 + 2 + 4 + 7 = 14 \quad \text{(not divisible by 3)} \]
- Check divisibility by \( 5 \) (not applicable as it does not end in \( 0 \) or \( 5 \)).
- Check divisibility by \( 7 \): \[ 1,247 \div 7 = 178.142857... \quad \text{(not divisible)} \]
- Check divisibility by \( 11 \): \[ 1,247 \div 11 = 113.363636... \quad \text{(not divisible)} \]
- Check divisibility by \( 13 \): \[ 1,247 \div 13 = 96.6923077... \quad \text{(not divisible)} \]
- Check divisibility by \( 17 \): \[ 1,247 \div 17 = 73.3529411... \quad \text{(not divisible)} \]
- Finally, check divisibility by \( 19 \): \[ 1,247 \div 19 = 65.1052631... \quad \text{(not divisible)} \]
- We find that \( 1,247 = 19 \times 67 \) after testing.
So the full factorization of \( 9,976 \) is: \[ 9,976 = 2^3 \times 19 \times 67 \]
Since \( 15 \) does not share any common factors with \( 9,976 \), the greatest common divisor (gcd) of \( 15 \) and \( 9,976 \) is \( 1 \).
Thus, the fraction \( \frac{15}{9,976} \) is already in its simplest form.
Final answer: \[ \frac{15}{9,976} \]