Reduce the fraction 3596 / 4292 to its lowest terms by cancelling the greatest common divisor of the numerator and denominator.

Introduction / Context:
This question checks your ability to simplify fractions by dividing both the numerator and the denominator by their greatest common divisor. Simplifying fractions is an essential arithmetic skill and appears frequently in aptitude tests, data interpretation sets, and number systems questions. A fully simplified fraction has no common factor greater than 1 between its numerator and denominator.

Given Data / Assumptions:
- The original fraction is 3596 / 4292.
- We assume both numbers are positive integers.
- We must find an equivalent fraction in lowest terms that matches one of the given options.

Concept / Approach:
To reduce a fraction to lowest terms, we find the greatest common divisor of the numerator and denominator and divide both by this number. The Euclidean algorithm or systematic factorisation can be used, but often noticing divisibility patterns is enough. If the result matches one of the answer choices, that is the simplified form.

Step-by-Step Solution:
Step 1: Check basic divisibility. Both 3596 and 4292 are even numbers, so they are divisible by 2. Step 2: Divide both numbers by 2: 3596 / 2 = 1798 and 4292 / 2 = 2146. The fraction becomes 1798 / 2146. Step 3: Both 1798 and 2146 are still even, so divide again by 2: 1798 / 2 = 899 and 2146 / 2 = 1073. Now the fraction is 899 / 1073. Step 4: Check if 899 and 1073 have a common factor. By inspection or using the Euclidean algorithm, we find that 899 = 29 × 31 and 1073 = 29 × 37. Step 5: Divide numerator and denominator by the common factor 29 to get 31 / 37. Step 6: The numbers 31 and 37 are prime and share no common factor other than 1, so 31 / 37 is the fraction in lowest terms.
Verification / Alternative check:
Multiply the simplified fraction back by the common factor to confirm. 31 × 116 = 3596 and 37 × 116 = 4292, where 116 is 4 × 29. This shows that 31 / 37 and 3596 / 4292 are equivalent. Also, since 31 and 37 are consecutive odd primes, there is no further simplification possible.

Why Other Options Are Wrong:
Option a, 29/37, would correspond to a numerator 29 × 116 = 3364, not 3596.
Option b, 17/43, and option d, 19/23, are unrelated ratios and do not give the same decimal value as the original fraction.
Option e, 29/31, also does not equal 3596 / 4292 when cross multiplied and therefore cannot be the simplified form.

Common Pitfalls:
A common mistake is to stop after dividing by a small factor such as 2 or 4 and assume the result is fully simplified. Another error is incorrect factorisation of the numbers, which leads to an incorrect reduced fraction. Always check whether the resulting numerator and denominator still share a common factor, especially when the starting numbers are large. Prime factorisation or the Euclidean algorithm can help avoid such errors.

Final Answer:
The fraction 3596 / 4292 in lowest terms is 31 / 37.