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Number Theory Level 4

Find the sum of all positive integers x x such that

x + 3 ∣ x 3 + x 2 + x + 1 x + 3 \mid x^3 + x^2 + x + 1


The answer is 27.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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2 solutions

Aaaaa Bbbbb
Aug 18, 2014

x 3 + x 2 + x + 1 = ( x 2 − 2 x + 7 ) × ( x + 3 ) − 20 x^3+x^2+x+1=(x^2-2x+7) \times (x+3) - 20 ⇒ ( x + 3 ) ∣ 20 ⇒ x = 1 ∣ 2 ∣ 7 ∣ 17 \Rightarrow (x+3) | 20 \Rightarrow x = 1|2|7|17 R e s = 27 Res = \boxed{27}

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R e s Res ? Abbreviation of R e s u l t Result ?

mathh mathh - 6 years, 9 months ago

I see that the question has been edited from it's original version.

Those who previously answered 1 were marked correct. The correct answer is now 27.

Calvin Lin Staff - 6 years, 9 months ago

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Yes, I completely muddled it all up! I was doing it near midnight so I wasn't really paying attention.

Sharky Kesa - 6 years, 9 months ago

I also arrived at this answer with a similar solution. To check, I wrote a small program to check solutions within the range (1,100000). The solutions were 1, 2, 7, 17 and 32775! Any ideas, why this is the case?

Soham Karwa - 6 years, 9 months ago

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According to my calculator, if x = 32775 x = 32775 , x 3 + x 2 + x + 1 x + 3 = 1074135081 + 16379 16389 \displaystyle \frac{x^{3}+x^{2}+x+1}{x+3} = 1074135081 + \displaystyle \frac{16379}{16389}

That's really close to whole number though.

Samuraiwarm Tsunayoshi - 6 years, 9 months ago

That's strange. Make sure that your program doesn't have any rounding errors, as 32775 is a rather big number.

Daniel Liu - 6 years, 9 months ago
Akash Deep
Aug 22, 2014

s a y , f ( x ) = x 3 + x 2 + x + 1 f ( x ) = ( x + 3 ) ( x 2 − 2 x + 7 ) − 20 n o w , w e n e e d t o f i n d x f o r w h i c h x + 3 ∣ 20 s o t h e o n l y v a l u e s a r e a r e 2 , 1 , 7 , 17 s u m = 27 say,\quad f(x)\quad =\quad { x }^{ 3 }+{ x }^{ 2 }+x+1\\ f(x)\quad =\quad (x+3)({ x }^{ 2 }-2x+7)\quad -\quad 20\\ now,\quad we\quad need\quad to\quad find\quad x\quad for\quad which\quad \\ x+3\quad |\quad 20\\ so\quad the\quad only\quad values\quad are\quad are\quad 2,1,7,17\\ sum\quad =\quad 27

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