Find the number of quadratic polynomials, , which satisfy the following conditions:
(a)
are distinct;
(b)
, and
(c)
divides
.
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Condition (c) is equivalent to a + c = b . For a given value of b , there are exactly b − 1 pairs of positive integers ( a , c ) summing to b , but if b is even, we have to throw out a = c = b / 2 , so there are b − 2 solutions for even b and b − 1 solutions for odd b . So the total number of solutions is 0 + 0 + 2 + 2 + 4 + 4 + ⋯ + 1 9 9 6 + 1 9 9 6 + 1 9 9 8 = 4 ( 1 + 2 + ⋯ + 9 9 8 ) + 1 9 9 8 = 9 9 8 ⋅ 9 9 9 ⋅ 2 + 1 9 9 8 = 2 ⋅ 9 9 9 2 = 1 9 9 6 0 0 2 .