Find the last three digits of the sum of all positive integers n<1000, such that the polynomial fn(x)=x4+n is a product of two non-constant polynomials with integer coefficients.
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Is the ans (4 64 324)=392 ? I don't think, that these three values are the only values of 'n'. Because, the question is to find the last three digits of the sum. But here, the sum itself is a three digit number, so, I think, there must be some other values of 'n'. In my solution, I found that, n must be in the form 4*(a^4) [where 'a' is a positive integer]. In this way, we have, only three values.
Shame on you Keshav!!! Cheating on Brilliant!!! This site is meant to be a platform to practice your own skills, not to copy paste the questions and get free answers and then get incentives without effort. So either play fair and be honest or leave this site. People like you are shame to the Brilliant community. And to the others, please give the answer to this problem after Monday 10/6/2013, so that this cheat doesn't get the opportunity to cheat.
Debjit what did you do? This problem is an open problem in this website:- (h)(t)(t)(p)(:)(/)(/)(brilliant)(dot)(org) [just remove the brackets and place '.' in place of (dot)]. If you register there you'll see it is a great side which offers weekly problems to solve for free and gives incentives in return. But some greedy cheats like Keshav always op for shortcuts and post the problems here. This problem was worth more than 100 points, and this cheat has got them without any effort.
To find the last three digits of the sum of all positive integers n < 1000, such that the polynomial f_n(x) = x^4 + n is a product of two non-constant polynomials with integer coefficients, we need to determine which values of n satisfy this condition.
Let's break down the problem into steps:
1. Start by considering the factorization of the polynomial f_n(x) = x^4 + n. If it is a product of two non-constant polynomials with integer coefficients, it means that x^4 + n can be factored as (x^2 + ax + b)(x^2 + cx + d), where a, b, c, and d are integers.
2. Multiply out this equation to get the expanded form: x^4 + (a + c)x^3 + (ac + b + d)x^2 + (ad + bc)x + bd.
3. Equating the coefficients of the expanded form with the corresponding coefficients of the original polynomial x^4 + n, we have:
(a + c) = 0,
(ac + b + d) = 0,
(ad + bc) = 0,
bd = n.
4. From the first equation, we can conclude that a = -c.
5. Substituting this value into the second equation, we get b + d = -ac.
6. From the fourth equation, we have bd = n. Since n must be positive, b and d must have the same sign.
7. Based on the six possible cases for the signs of b and d (++, +-, -+, --), we can determine the values of a, b, c, and d for each case.
8. For each case, we can compute the sum of the values of n that satisfy the conditions and find the last three digits of that sum.
By following these steps, you should be able to determine the last three digits of the sum of all positive integers n < 1000, such that the polynomial f_n(x) = x^4 + n is a product of two non-constant polynomials with integer coefficients.