55rd Ukrainian National Mathematical Olympiad - Fourth Round

Question

a) Determine whether there exist positive integer numbers

a_{1}, a_{2}, ..., a_{2015}

such that any two of them are co-prime and

a_{1} a_{2} ... a_{2015} - 1

is a product of two consequent odd numbers?

b) Determine whether there exist positive integer numbers

a_{1}, a_{2}, ..., a_{2015}

such that: any two of them are co-prime and

a_{1} a_{2} ... a_{2015} - 1

is a product of two consequent even numbers?

Accepted Answer

a) Let a_1 = p_1^2, a_2 = p_2^2, ..., a_2015 = p_2015^2, where p_1 = 2, p_2, ..., p_2015 are the first 2015 prime numbers. It is clear that every two of these numbers are co-prime and a_1a_2...a_2015-1 = (p_1p_2...p_2015-1)(p_1p_2...p_2015+1) is a product of two consequent odd numbers. b) Let a_1 = p_1^2, a_2 = p_2^2, ..., a_2015 = p_2015^2, where p_1 = 3, p_2 = 5, ..., p_2015 are the first 2015 odd prime numbers. It is clear that every two of these numbers are co-prime and a_1a_2...a_2015-1 = (p_1p_2...p_2015-1)(p_1p_2...p_2015+1) is a product of two consequent even numbers.