Let's reduce each of these fractions to their simplest form by identifying and dividing by their greatest common divisor (GCD).
### Step-by-Step Solutions
a. [tex]\(\frac{8}{36}\)[/tex]
1. Identify the GCD of 8 and 36:
- Factors of 8: [tex]\(1, 2, 4, 8\)[/tex]
- Factors of 36: [tex]\(1, 2, 3, 4, 6, 9, 12, 18, 36\)[/tex]
- The GCD is 4.
2. Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{8 \div 4}{36 \div 4} = \frac{2}{9}
\][/tex]
Thus, [tex]\(\frac{8}{36}\)[/tex] simplifies to [tex]\(\frac{2}{9}\)[/tex].
b. [tex]\(\frac{98}{147}\)[/tex]
1. Identify the GCD of 98 and 147:
- Factors of 98: [tex]\(1, 2, 7, 14, 49, 98\)[/tex]
- Factors of 147: [tex]\(1, 3, 7, 21, 49, 147\)[/tex]
- The GCD is 49.
2. Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{98 \div 49}{147 \div 49} = \frac{2}{3}
\][/tex]
Thus, [tex]\(\frac{98}{147}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex].
c. [tex]\(\frac{75}{165}\)[/tex]
1. Identify the GCD of 75 and 165:
- Factors of 75: [tex]\(1, 3, 5, 15, 25, 75\)[/tex]
- Factors of 165: [tex]\(1, 3, 5, 11, 15, 33, 55, 165\)[/tex]
- The GCD is 15.
2. Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{75 \div 15}{165 \div 15} = \frac{5}{11}
\][/tex]
Thus, [tex]\(\frac{75}{165}\)[/tex] simplifies to [tex]\(\frac{5}{11}\)[/tex].
d. [tex]\(\frac{105}{231}\)[/tex]
1. Identify the GCD of 105 and 231:
- Factors of 105: [tex]\(1, 3, 5, 7, 15, 21, 35, 105\)[/tex]
- Factors of 231: [tex]\(1, 3, 7, 11, 21, 33, 77, 231\)[/tex]
- The GCD is 21.
2. Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{105 \div 21}{231 \div 21} = \frac{5}{11}
\][/tex]
Thus, [tex]\(\frac{105}{231}\)[/tex] simplifies to [tex]\(\frac{5}{11}\)[/tex].
### Summary of Simplified Fractions:
a. [tex]\(\frac{8}{36} = \frac{2}{9}\)[/tex]
b. [tex]\(\frac{98}{147} = \frac{2}{3}\)[/tex]
c. [tex]\(\frac{75}{165} = \frac{5}{11}\)[/tex]
d. [tex]\(\frac{105}{231} = \frac{5}{11}\)[/tex]
