1.3 Show through factorizing that 899 is not a prime numb

1. **Understanding the Problem** We need to show that \(899\) is not a prime number by factorizing it into two or more factors greater than \(1\). 2. **Checking Divisibility** A prime number has no divisors other than \(1\) and itself. To test if \(899\) can be factorized, we first check divisibility by small prime numbers. Note that the square root of \(899\) is approximately \[ \sqrt{899} \approx 30. \] Therefore, it is sufficient to check for factors less than or equal to \(30\). 3. **Testing Potential Factors** We test divisibility by some of the small primes: - \(899 \div 2\) is not an integer (since \(899\) is odd). - \(899 \div 3\): The sum of the digits is \(8 + 9 + 9 = 26\), which is not divisible by \(3\). - \(899 \div 5\) is not an integer (since \(899\) does not end in \(0\) or \(5\)). - Continue testing with primes like \(7, 11, 13, 17, 19,\) and \(23\) until a divisor is found. 4. **Finding the Correct Factors** When testing \(29\): \[ 899 \div 29 = 31 \] Since \(31\) is an integer and also a prime number, we have found a successful factorization. 5. **Conclusion** We have factorized \(899\) as: \[ 899 = 29 \times 31. \] Since both \(29\) and \(31\) are prime numbers, \(899\) is not a prime number.