Algebra
# Arithmetic Progressions

Let \(x^3+ax^2+bx+c\) be a polynomial with positive integer coefficients which are in arithmetic progression in that order.

Determine the maximum possible number of integer roots of the polynomial.

What is the largest positive integer that must be a divisor of the sum of an arithmetic sequence with 90 integer terms?

Try Part II

\[ \begin{array}{llllll} A_1 : & 2, & 9, & 16, \ldots , & 2 + (1000-1) \times 7 \\ A_2: & 3, & 12 , & 21, \ldots, & 3 + (1000-1) \times 9? \\ \end{array} \]

How many integers appear in both of the following arithmetic progressions above?

**Details and assumptions**

Since 2 appears in \(A_1\) but not in \(A_2\), it does not appear in both of the arithmetic progressions.

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