Quote by buzzmath
the problem i'm encountering is we know (xj)^(n1)/(pj) is incongruent to 1(modn) but (xj)^(n1)/(pi) may not be incongruent to 1(modn) for an i≠j

You can't get around this problem. If n is prime, just take xj to be an element with order pj^aj.
Fortunately though, you don't need to. Just consider the condition for one value of j at a time. What does it tell you about the power of pj that appears in phi(n)?