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Answer :
Sure! Let's do a step-by-step breakdown of the prime factorization of 198.
1. Start with the number 198.
2. Check for the smallest prime factor:
- 198 is even, so it is divisible by 2.
- Divide 198 by 2:
[tex]\[
198 \div 2 = 99
\][/tex]
- So, one factor is 2.
3. Now consider 99:
- 99 is not even, so move to the next smallest prime number, which is 3.
- Check if 99 is divisible by 3. The sum of the digits of 99 is 9 (since [tex]\( 9 + 9 = 18 \)[/tex] and 18 is divisible by 3), so 99 is also divisible by 3.
- Divide 99 by 3:
[tex]\[
99 \div 3 = 33
\][/tex]
- So, another factor is 3.
4. Consider 33:
- Again, check if 33 is divisible by 3. The sum of the digits of 33 is 6 (since [tex]\( 3 + 3 = 6 \)[/tex]), and 6 is divisible by 3.
- Divide 33 by 3:
[tex]\[
33 \div 3 = 11
\][/tex]
- So, another factor is 3.
5. Finally, consider 11:
- 11 is a prime number, which means it cannot be divided by any other number except itself and 1.
- So, the last factor is 11.
6. Combine all the prime factors:
- We listed all the prime factors: 2, 3, 3, and 11.
So, when you put them all together, the prime factorization of 198 is:
[tex]\[
198 = 2 \times 3^2 \times 11
\][/tex]
1. Start with the number 198.
2. Check for the smallest prime factor:
- 198 is even, so it is divisible by 2.
- Divide 198 by 2:
[tex]\[
198 \div 2 = 99
\][/tex]
- So, one factor is 2.
3. Now consider 99:
- 99 is not even, so move to the next smallest prime number, which is 3.
- Check if 99 is divisible by 3. The sum of the digits of 99 is 9 (since [tex]\( 9 + 9 = 18 \)[/tex] and 18 is divisible by 3), so 99 is also divisible by 3.
- Divide 99 by 3:
[tex]\[
99 \div 3 = 33
\][/tex]
- So, another factor is 3.
4. Consider 33:
- Again, check if 33 is divisible by 3. The sum of the digits of 33 is 6 (since [tex]\( 3 + 3 = 6 \)[/tex]), and 6 is divisible by 3.
- Divide 33 by 3:
[tex]\[
33 \div 3 = 11
\][/tex]
- So, another factor is 3.
5. Finally, consider 11:
- 11 is a prime number, which means it cannot be divided by any other number except itself and 1.
- So, the last factor is 11.
6. Combine all the prime factors:
- We listed all the prime factors: 2, 3, 3, and 11.
So, when you put them all together, the prime factorization of 198 is:
[tex]\[
198 = 2 \times 3^2 \times 11
\][/tex]
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