In this article, we solve one of the oldest and most celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there is no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all even perfect numbers. We answer the question of whether there exists an odd perfect number in the negative by proving a theorem asserting that the existence of such a number would lead to contradictions (proof by reductio ad absurdum). Somewhat remarkably, perhaps, this result is proved using only elementary methods. Hence, the popular conjecture that odd perfect numbers do not exist, no matter how large these numbers might be, is confirmed to be correct. Thus, one of the oldest and most celebrated questions in mathematics has now a definitive answer.