Answer :
To find the prime factorization of the composite number 196, you can follow these steps:
1. Start by testing smaller prime numbers:
- Check if 196 is divisible by 2 (since 196 is even).
- [tex]\( 196 \div 2 = 98 \)[/tex]. So, 2 is a factor, and we write it as:
[tex]\[ 196 = 2 \times 98 \][/tex]
2. Check 98 for further factors:
- 98 is also even, so it can be divided by 2 again.
- [tex]\( 98 \div 2 = 49 \)[/tex]. Now we have:
[tex]\[ 196 = 2 \times 2 \times 49 \][/tex]
- This can be written as:
[tex]\[ 196 = 2^2 \times 49 \][/tex]
3. Factorize 49:
- 49 is not even; try the next smallest prime, which is 3 (49 is not divisible by 3 since [tex]\( 4 + 9 = 13 \)[/tex], which is not divisible by 3).
- Try 5 (49 is not divisible by 5, as it does not end in 0 or 5).
- Try 7 (7 × 7 = 49), so it is:
[tex]\[ 49 = 7 \times 7 \][/tex]
- We can rewrite 196 as:
[tex]\[ 196 = 2^2 \times 7 \times 7 \][/tex]
or
[tex]\[ 196 = 2^2 \times 7^2 \][/tex]
So the prime factorization of the number 196 is:
[tex]\[ 196 = 2^2 \times 7^2 \][/tex]
This means that 196 is made up of the prime factors 2 and 7, each raised to the power of 2.
1. Start by testing smaller prime numbers:
- Check if 196 is divisible by 2 (since 196 is even).
- [tex]\( 196 \div 2 = 98 \)[/tex]. So, 2 is a factor, and we write it as:
[tex]\[ 196 = 2 \times 98 \][/tex]
2. Check 98 for further factors:
- 98 is also even, so it can be divided by 2 again.
- [tex]\( 98 \div 2 = 49 \)[/tex]. Now we have:
[tex]\[ 196 = 2 \times 2 \times 49 \][/tex]
- This can be written as:
[tex]\[ 196 = 2^2 \times 49 \][/tex]
3. Factorize 49:
- 49 is not even; try the next smallest prime, which is 3 (49 is not divisible by 3 since [tex]\( 4 + 9 = 13 \)[/tex], which is not divisible by 3).
- Try 5 (49 is not divisible by 5, as it does not end in 0 or 5).
- Try 7 (7 × 7 = 49), so it is:
[tex]\[ 49 = 7 \times 7 \][/tex]
- We can rewrite 196 as:
[tex]\[ 196 = 2^2 \times 7 \times 7 \][/tex]
or
[tex]\[ 196 = 2^2 \times 7^2 \][/tex]
So the prime factorization of the number 196 is:
[tex]\[ 196 = 2^2 \times 7^2 \][/tex]
This means that 196 is made up of the prime factors 2 and 7, each raised to the power of 2.
