We need to determine which of the given fractions is in the simplest form. A fraction is in simplest form if the greatest common divisor (GCD) of its numerator and denominator is 1.
Let's check each fraction:
1. Fraction: [tex]\(\frac{39}{65}\)[/tex]
- Find the GCD of 39 and 65.
- Factors of 39: 1, 3, 13, 39
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{39}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[
\frac{39 \div 13}{65 \div 13} = \frac{3}{5}
\][/tex]
- Hence, [tex]\(\frac{39}{65}\)[/tex] is not in simplest form.
2. Fraction: [tex]\(\frac{45}{65}\)[/tex]
- Find the GCD of 45 and 65.
- Factors of 45: 1, 3, 5, 9, 15, 45
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{45}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[
\frac{45 \div 5}{65 \div 5} = \frac{9}{13}
\][/tex]
- Hence, [tex]\(\frac{45}{65}\)[/tex] is not in simplest form.
3. Fraction: [tex]\(\frac{46}{65}\)[/tex]
- Find the GCD of 46 and 65.
- Factors of 46: 1, 2, 23, 46
- Factors of 65: 1, 5, 13, 65
- The common factor is 1.
- The GCD is 1.
- Therefore, [tex]\(\frac{46}{65}\)[/tex] is already in simplest form since we cannot divide it any further. This fraction doesn't need any simplification.
4. Fraction: [tex]\(\frac{52}{65}\)[/tex]
- Find the GCD of 52 and 65.
- Factors of 52: 1, 2, 4, 13, 26, 52
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 13.
- The GCD is 13.
- Therefore, [tex]\(\frac{52}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 13:
[tex]\[
\frac{52 \div 13}{65 \div 13} = \frac{4}{5}
\][/tex]
- Hence, [tex]\(\frac{52}{65}\)[/tex] is not in simplest form.
5. Fraction: [tex]\(\frac{55}{65}\)[/tex]
- Find the GCD of 55 and 65.
- Factors of 55: 1, 5, 11, 55
- Factors of 65: 1, 5, 13, 65
- The common factors are 1 and 5.
- The GCD is 5.
- Therefore, [tex]\(\frac{55}{65}\)[/tex] can be simplified by dividing both the numerator and the denominator by 5:
[tex]\[
\frac{55 \div 5}{65 \div 5} = \frac{11}{13}
\][/tex]
- Hence, [tex]\(\frac{55}{65}\)[/tex] is not in simplest form.
Among the given fractions, the fraction [tex]\(\frac{46}{65}\)[/tex] is already in simplest form. Thus, the fraction in simplest form is:
[tex]\[
\boxed{\frac{46}{65}}
\][/tex]
