For a prime http://alt1.artofproblemsolving.com/...5310b19759.gif and a given integer http://alt2.artofproblemsolving.com/...00c274bdaa.gif let http://alt1.artofproblemsolving.com/...6bdaafbbf1.gif denote the exponent of http://alt1.artofproblemsolving.com/...5310b19759.gif in the prime factorisation of http://alt2.artofproblemsolving.com/...c31712f5af.gif. Given http://alt1.artofproblemsolving.com/...d32c220eb7.gif and http://alt1.artofproblemsolving.com/...b40c51d10a.gif a set of http://alt1.artofproblemsolving.com/...c1970cff0c.gif primes, show that there are infinitely many positive integers http://alt2.artofproblemsolving.com/...00c274bdaa.gif such that http://alt1.artofproblemsolving.com/...944859d499.gif for all http://alt2.artofproblemsolving.com/...c1c25e73fb.gif.