To simplify the fraction \( \frac{1287}{1225} \), we can start by finding the greatest common divisor (GCD) of the two numbers.
First, we list the prime factorization of each number:
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Finding the prime factorization of 1225:
- 1225 is \( 35 \times 35 \) (since \( 35^2 = 1225 \)).
- Further factor \( 35 = 5 \times 7 \).
- So, \( 1225 = 5^2 \times 7^2 \).
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Finding the prime factorization of 1287:
- We can divide by 3 (sum of digits \( 1 + 2 + 8 + 7 = 18 \), which is divisible by 3): \( 1287 \div 3 = 429 \).
- Next, we factor 429. The sum of the digits \( 4 + 2 + 9 = 15 \) is divisible by 3: \( 429 \div 3 = 143 \).
- Finally, we factor 143. The factors of 143 are \( 11 \times 13 \) (since both 11 and 13 are prime numbers).
Now, we have: \[ 1287 = 3^2 \times 11 \times 13, \] \[ 1225 = 5^2 \times 7^2. \]
Now, since there are no common factors between \( 1287 \) (which has prime factors 3, 11, and 13) and \( 1225 \) (which has prime factors 5 and 7), the GCD is \( 1 \).
Thus, \( \frac{1287}{1225} \) is already in its simplest form. Therefore, the simplified form of \( 1287 \div 1225 \) is:
\[ \frac{1287}{1225}. \]
There is no further simplification possible.
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