An interesting problem in numbers (Q790868)

Let \(N_ v\) and \(M_ u\) be natural numbers with v and u digits, respectively, in the decimal system, where initial zeros are allowed. The paper gives a construction to find integers v, \(N_ v\) and \(M_ u\) such that for given u, a and b (a and b relatively prime) the equality \(b (10^ u N_ v+M_ u)=a (10^ v M_ u+N_ v)\) is satisfied.