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Integer Identification (Posted on 2015-02-14) Difficulty: 3 of 5
Find all integer values of p such that x3 + 2x + p divides:
x12 - x11 + 3x10 + 11x3 - x2 + 23x + 30.

No Solution Yet Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Narrowing the Field | Comment 1 of 3
I don't feel like doing long division right now, so let's just try some values.

a) If x = 0, then we have p divides 30
b) If x = 1, then we have p+3 divides 66
c) If x = -1, then we have p-3 divides 0

(unless I have done the math wrong)

(c) provides no information.
(a) means that p is either 1,2,3,5, 6, 10,15, 30, -1, -2, -3,-5, -6, -10, -15 or -30.  
But p+3 must divide 66, so the only viable candidates are 3, 30, -2 and -6.

Well, that's encouraging.  Let's try a few more values.

d) If x = 2, then we have p+12 divides 5280
e) If x = -2, then we have p-4 divides 9108

(more opportunities for math errors)

Based on (d), we can eliminate p = 30
Based on (e), we can eliminate p = -6

So the only viable candidates are p = 3 and p = -2.
Of course, I haven't proven that either one works for all x.  
But the long division is more viable now.  Perhaps I'll finish it off after grocery shopping. 

Edited on February 15, 2015, 11:09 am
  Posted by Steve Herman on 2015-02-14 09:17:21

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