Enter the prime factorization of 198.
The task is to find the prime factorization of the integer 198. A prime factorization represents a number as a product of its prime factors. Since 198 is an even number, it is divisible by the smallest prime, 2. Dividing 198 by 2, we get 99. Now we need to factor 99, which is not divisible by 2. The next prime number is 3. Since the sum of the digits of 99 is \(9+9=18\), which is divisible by 3, 99 is divisible by 3. Dividing 99 by 3 gives 33. Dividing 33 by 3 gives 11. Finally, 11 is a prime number as it has no divisors other than 1 and itself.
Collecting all the identified prime factors (2, 3, 3, and 11), we express 198 as the product: \(198 = 2 \times 3 \times 3 \times 11\). Using exponents for repeated factors, this simplifies to \(2 \times 3^2 \times 11\). Thus, the prime factorization of 198 is \(\boxed{2 \times 3^2 \times 11}\).
Prime factorization is essential in cryptography, where large numbers are factored to secure digital communications. In everyday scenarios, finding the prime factors of a number can help in simplifying complex fractions or determining the least common multiple of several numbers, which is useful when organizing schedules or calculating synchronized events. Knowing that \(198 = 2 \times 3^2 \times 11\) allows us to quickly find all its divisors: 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, and 198.