Questions: Find the prime factorization of 245.

Transcript text: Find the prime factorization of 245.

Solution

Solution Steps

To find the prime factorization of 245, we need to divide the number by the smallest prime number (starting from 2) and continue dividing the quotient by the smallest prime number until the quotient itself is a prime number.

Step 1: Identify the smallest prime factor

To find the prime factorization of 245, we start by identifying the smallest prime factor. The smallest prime number is 2, but 245 is odd, so it is not divisible by 2. The next smallest prime number is 3, but the sum of the digits of 245 (2 + 4 + 5 = 11) is not divisible by 3. The next smallest prime number is 5, and since 245 ends in 5, it is divisible by 5.

Step 2: Divide by the smallest prime factor

We divide 245 by 5: \[ 245 \div 5 = 49 \] So, 5 is a prime factor of 245.

Step 3: Repeat the process with the quotient

Next, we need to factorize 49. The smallest prime number is 2, but 49 is odd, so it is not divisible by 2. The next smallest prime number is 3, but 49 is not divisible by 3. The next smallest prime number is 5, but 49 is not divisible by 5. The next smallest prime number is 7, and since \(49 = 7 \times 7\), 7 is a prime factor of 49.

Step 4: Continue until the quotient is a prime number

Since \(49 = 7 \times 7\), we have: \[ 49 \div 7 = 7 \] So, 7 is a prime factor of 49, and we are left with 7, which is a prime number.