To solve the problem of finding the value of [tex]\( q \)[/tex] in the prime factorization of 1176 given by [tex]\( 1176 = 2^p \times 3^q \times 7^r \)[/tex], we can proceed with the following steps:
1. Prime Factorization of 1176:
We want to factorize 1176 into its prime factors.
2. Divide by 2:
- [tex]\( 1176 \div 2 = 588 \)[/tex]
- [tex]\( 588 \div 2 = 294 \)[/tex]
- [tex]\( 294 \div 2 = 147 \)[/tex]
After these divisions, 147 is no longer divisible by 2. So, the power [tex]\( p \)[/tex] of 2 is 3.
3. Divide by 3:
- [tex]\( 147 \div 3 = 49 \)[/tex]
After this division, 49 is no longer divisible by 3. So, the power [tex]\( q \)[/tex] of 3 is 1.
4. Divide by 7:
- [tex]\( 49 \div 7 = 7 \)[/tex]
- [tex]\( 7 \div 7 = 1 \)[/tex]
After these divisions, the power [tex]\( r \)[/tex] of 7 is 2.
From the factorization, we obtain:
[tex]\[ 1176 = 2^3 \times 3^1 \times 7^2 \][/tex]
Thus, the value of [tex]\( q \)[/tex] is [tex]\( 1 \)[/tex].
So, the correct option is:
c) 1
