1988 AIME Problems/Problem 13

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Problem

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$.

Solution

Solution 1

Let's work backwards! Let $F(x) = ax^{17} + bx^{16} + 1$ and let $P(x)$ be the polynomial such that $P(x)(x^2 - x - 1) = F(x)$.

First, it's kinda obvious that the constant term of $P(x)$ must be $- 1$. Now, we have $(x^2 - x - 1)(c_1x^{15} + c_2x^{14} + \cdots + c_{15}x - 1)$, where $c_{15}$ is some random coefficient. However, since $F(x)$ has no $x$ term, it must be true that $c_{15} = - 1$.

Let's find $c_{14}$ now. Notice that all we care about in finding $c_{14}$ is that $(x^2 - x - 1)(\cdots + c_{14}x^2 + x - 1) = \text{something} + 0x^2 + \text{something}$. Therefore, $c_{14} = - 2$. Undergoing a similar process, $c_{13} = 3$, $c_{12} = - 5$, $c_{11} = 8$, and we see a nice pattern. The coefficients of $P(x)$ are just the Fibonacci sequence with alternating signs! Therefore, $a = c_1 = F_{16}$, where $F_{16}$ denotes the 16th Fibonnaci number and $a = 987$.

Solution 2

Let $F_n$ represent the $n$th number in the Fibonacci sequence. Therefore,

$x^2 - x - 1 = 0\Longrightarrow x^n = F_n(x),\ n\in N\Longrightarrow x^{n + 2} = F_{n + 1}\cdot x + F_n,\ n\in N\ .$

The above uses the similarity between the Fibonacci recursive definition, $F_{n+2} - F_{n+1} - F_n = 0$, and the polynomial $x^2 - x - 1 = 0$.

$0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1\Longrightarrow$

$(aF_{17} + bF_{16})\cdot x + (aF_{16} + bF_{15} + 1) = 0,\ x\not\in Q\Longrightarrow$

$aF_{17} + bF_{16} = 0$ and $aF_{16} + bF_{15} + 1 = 0\Longrightarrow$

$a = F_{16},\ b = - F_{17}\Longrightarrow \boxed {a = 987,\ b = - 1597}\ .$

Solution 3

We can long divide and search for a pattern; then the remainder would be set to zero to solve for $a$. Writing out a few examples quickly shows us that the remainders after each subtraction follow the Fibonacci sequence. Carrying out this pattern, we find that the remainder is $(F_{16} + F_{17}a)x + F_{15}b + F_{16}a + 1 = 0$. Since the coefficient of $x$ must be zero, this gives us two equations, $F_{16}b + F_{17}a = 0$ and $F_{15}b + F_{16}a + 1 = 0$. Solving these two as above, we get that $a = 987$.

There are various similar solutions which yield the same pattern, such as repeated substitution of $x^2 = x + 1$ into the larger polynomial.

Solution 4

The roots of $x^2-x-1$ are the golden ratio and 1-(the golden ratio). Call these $g$ and $1-g$. These two must also be roots of $ax^{17}+bx^{16}+1$. Thus, we have two equations: $ag^{17}+bg^{16}+1=0$ and $a(1-g)^{17}+b(1-g)^{16}+1=0$. Subtract these two and divide by $\sqrt{5}$ to get $a(g^{17}-(1-g)^{17})/\sqrt{5}+b(g^{16}-(1-g)^{16})/\sqrt{5}=0$. But the formula for the nth fibonacci number is $(g^n-(1-g)^n)/\sqrt{5})$ (You may want to research this). Thus, we have $1597a+987b=0$, so since a must be positive, we can guess it equals 987.

See also

1988 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions