36:39 what is the simplest form
The Correct Answer and Explanation is :
To simplify the fraction ( \frac{36}{39} ), we first need to find the greatest common divisor (GCD) of the numerator (36) and the denominator (39).
Step 1: Find the GCD
- Prime Factorization:
- The prime factorization of 36 is ( 2^2 \times 3^2 ) (since ( 36 = 2 \times 2 \times 3 \times 3 )).
- The prime factorization of 39 is ( 3 \times 13 ) (since ( 39 = 3 \times 13 )).
- Identify Common Factors:
- The only common prime factor between 36 and 39 is 3.
- Greatest Common Divisor:
- Thus, the GCD of 36 and 39 is 3.
Step 2: Simplify the Fraction
To simplify ( \frac{36}{39} ), we divide both the numerator and the denominator by their GCD (3):
[
\frac{36 \div 3}{39 \div 3} = \frac{12}{13}
]
Step 3: Verify the Simplification
Now, we can check if ( \frac{12}{13} ) is in its simplest form. The prime factorization of 12 is ( 2^2 \times 3 ), and 13 is a prime number. There are no common factors between 12 and 13, confirming that ( \frac{12}{13} ) is indeed in simplest form.
Conclusion
The simplest form of ( \frac{36}{39} ) is ( \frac{12}{13} ). Simplifying fractions involves finding the GCD of the numerator and denominator and dividing both by that number. This process ensures that the fraction is expressed in the most concise and understandable manner while retaining the same value. Understanding how to simplify fractions is crucial in various mathematical applications, from basic arithmetic to more complex algebra and beyond.