STEP II krub
1. Find the sum of those numbers between 1000 and 6000 every one of whose digits is one of the numbers 0, 2, 5 or 7, giving your answer as a product of primes.
2. (i) When the polynomial p(x) is divied by (x-1), (x-2), (x-3) the remailders are 3, 1, 5 respectively. Given that
p(x) = (x-1)(x-2)(x-3)q(x) + r(x),
where q(x) and r(x) are polynomials with r(x) having degree less than three, find r(x)
(ii) Find a polynomial P(x) of degree n+1, where n is a given positive integer, such that for each integer a satisfying 0<= a <= n , the remainder when Pn(x) is divided by (x-a) is a.
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