Number theory · Divisibility · Coprime integers · Fractions in lowest terms · Composite numbers
Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\), where \(n > 1\), be positive integers satisfying
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}. \]
Prove that \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) is a composite number.
Sign in to check answers, open hints, read the full solution, and track your progress. Statements are always free.
Serbian Republic Competition (Republicko takmicenje) 1995, high school grade I, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source