Are you smarter than an 8th grader? #3

Number Theory Level 2

This problem is part of the question set Mathematics in Anime . It appears in the anime Puella Magi Madoka Magica . Students attending the Math class are only 14 14 years old, yet they are expected to solve the following problem on the spot!

The integer solutions of the equation a 3 + a 2 − a b + b − 1 = 0 a^{3}+a^{2}-ab+b-1=0 can be expressed in ( a 1 , b 1 ) , ( a 2 , b 2 ) , … ( a n , b n ) (a_1, b_1), (a_2, b_2), \ldots (a_n, b_n) .

Find ∑ i = 1 n ( a i + b i ) \sum_{i=1}^n (a_i+b_i)

Image and translation credits: Puella Magi Wiki


The answer is 14.

Zk Lin by ZK LIn , 24, Malaysia

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2 solutions

Daniel Liu
Oct 11, 2015

The equation simplifies to a 3 + a 2 − 1 a − 1 = b \dfrac{a^3+a^2-1}{a-1}=b . Since b ∈ Z b\in\mathbb{Z} , we must have a 3 + a 2 − 1 a − 1 = a 2 + 2 a + 2 + 1 a − 1 ∈ Z ⟹ 1 a − 1 ∈ Z ⟹ a − 1 = ± 1 ⟹ a = 0 , 2 \dfrac{a^3+a^2-1}{a-1}=a^2+2a+2+\dfrac{1}{a-1}\in\mathbb{Z}\implies \dfrac{1}{a-1}\in\mathbb{Z}\implies a-1=\pm 1\implies a=0,2 These two values give b = 1 , 11 b=1, 11 respectively so our solutions are ( a , b ) = ( 0 , 1 ) , ( 2 , 11 ) (a,b)=(0,1), (2,11) and the answer is 0 + 1 + 2 + 11 = 14 0+1+2+11=\boxed{14}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

I'm a bit frustrated when I see the equation. But when I see what is written in the paper, I can easily solve using your way! Thanks for the paper given me some hints.

Tran Quoc Dat - 5 years, 2 months ago
Aareyan Manzoor
Oct 9, 2015

factor: a a 3 + a 2 − a b + b − 1 = 0 ⟹ a 3 − 1 + a 2 − 1 − b ( a − 1 ) = − 1 ⟹ ( a − 1 ) ( a 2 + a + 1 ) + ( a − 1 ) ( a + 1 ) − ( a − 1 ) b = − 1 ⟹ ( a − 1 ) ( a 2 + 2 a + 2 − b ) = − 1 \begin{array}{c}a a^3+a^2-ab+b-1=0\\ \Longrightarrow a^3-1+a^2-1-b(a-1)=-1\\ \Longrightarrow (a-1)(a^2+a+1)+(a-1)(a+1)-(a-1)b=-1\\ \Longrightarrow (a-1)(a^2+2a+2-b)=-1 \end{array} for a , b ∈ Z a,b \in Z , { a − 1 = 1 → a = 2 , 2 2 + 2 ∗ 2 + 2 − b = − 1 → b = 11 a − 1 = − 1 → a = 0 , 0 2 + 2 ∗ 0 + 2 − b = 1 → b = 1 \begin{cases} a-1=1\rightarrow a=2,2^2+2*2+2-b=-1\rightarrow b=11\\ a-1=-1\rightarrow a=0,0^2+2*0+2-b=1\rightarrow b=1\end{cases} so the answer is 2 + 11 + 0 + 1 = 14 2+11+0+1=\boxed{14} i'm both eight grader and 13 as of oct '15

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Oh look, same approach! Although, I'm already in grade 10 right now, so...? I dunno what that means.

Manuel Kahayon - 5 years, 3 months ago

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The topic of the problem :p

Aareyan Manzoor - 5 years, 3 months ago

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