To simplify \( \frac{1287}{1225} \), we can start by finding the greatest common divisor (GCD) of the two numbers.
First, let's factor both numbers:
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For 1287:
- Dividing by 3: \( 1287 \div 3 = 429 \)
- Dividing 429 by 3 again: \( 429 \div 3 = 143 \)
- Now factor 143: \( 143 = 11 \times 13 \) (since 11 and 13 are prime)
So, the prime factorization of 1287 is: \[ 1287 = 3^2 \times 11 \times 13 \]
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For 1225:
- Dividing by 5: \( 1225 \div 5 = 245 \)
- Dividing 245 by 5 again: \( 245 \div 5 = 49 \)
- We know that \( 49 = 7^2 \)
So, the prime factorization of 1225 is: \[ 1225 = 5^2 \times 7^2 \]
Now we check if there are any common factors:
- The factors of 1287 are \( 3^2, 11, 13 \)
- The factors of 1225 are \( 5^2, 7^2 \)
Since there are no common prime factors, the GCD is 1.
This means \( \frac{1287}{1225} \) is already in its simplest form.
Thus, the simplified form is: \[ \frac{1287}{1225} \]