algebra question very easy please see ITS COMPLETE QUESTION

algebra question very easy please see

ITS COMPLETE QUESTION

The algebraic sum of any number of irreducible fractions whose denominators are relatively prime to each other cannot be an integer. That is, if ged(a_i, b_i) = 1, i = I, ..., n, and gcd(b_i, b_j) -= 1 for i notequalto j, show that a_1 + b_1 + a_2/b_2 + ... + a_n/b_n

Solution

Let us first approach this problem with n= 2. Let us also assume that there is an integer p such that                  a₁ / b₁ + a₂ / b₂ = p. Then ( b₁a₂ + b₂a₁ )/ b₁ b₂ = p or, b₁a₂ + b₂a₁ = p( b₁b₂). Now, if the above statement is true, then , since b and b are relatively prime, a₁ has to be a multiple of b₁ and a₂ must be a multiple of b₂ which is incorrect/a contradiction. Hence there can not be an integer p such that a₁ / b₁ + a₂ / b₂ = p. This argument can be extended to n = 1,3,4,5,…. and can thus be generalised.