Difference between revisions of "2017 AIME I Problems/Problem 9"

Revision as of 22:28, 12 February 2018

Problem 9

Let $a_{10} = 10$ , and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$ . Find the least $n > 10$ such that $a_n$ is a multiple of $99$ .

Solution 1

Writing out the recursive statement for $a_n, a_{n-1}, \dots, a_{10}$ and summing them gives $a_n+\dots+a_{10}=100(a_{n-1}+\dots+a_{10})+n+\dots+10$ Which simplifies to $a_n=99(a_{n-1}+\dots+a_{10})+\frac{1}{2}(n+10)(n-9)$ Therefore, $a_n$ is divisible by 99 if and only if $\frac{1}{2}(n+10)(n-9)$ is divisible by 99, so $(n+10)(n-9)$ needs to be divisible by 9 and 11. Assume that $n+10$ is a multiple of 11. Writing out a few terms, $n=12, 23, 34, 45$ , we see that $n=45$ is the smallest $n$ that works in this case. Next, assume that $n-9$ is a multiple of 11. Writing out a few terms, $n=20, 31, 42, 53$ , we see that $n=53$ is the smallest $n$ that works in this case. The smallest $n$ is $\boxed{045}$ .

Note that we can also construct the solution using CRT by assuming either $11$ divides $n+10$ and $9$ divides $n-9$ , or $9$ divides $n+10$ and $11$ divides $n-9$ , and taking the smaller solution.

Solution 2

$a_n \equiv a_{n-1} + n \pmod {99}$ By looking at the first few terms, we can see that $a_n \equiv 10+11+12+ \dots + n \pmod {99}$ This implies $a_n \equiv \frac{n(n+1)}{2} - \frac{10*9}{2} \pmod {99}$ Since $a_n \equiv 0 \pmod {99}$ , we can rewrite the equivalence, and simplify $0 \equiv \frac{n(n+1)}{2} - \frac{10*9}{2} \pmod {99}$ $0 \equiv n(n+1) - 90 \pmod {99}$ $0 \equiv 4n^2+4n+36 \pmod {99}$ $0 \equiv (2n+1)^2+35 \pmod {99}$ $64 \equiv (2n+1)^2 \pmod {99}$ The only squares that are congruent to $64 \pmod {99}$ are $(\pm 8)^2$ and $(\pm 19)^2$ , so $2n+1 \equiv -8, 8, 19, \text{or } {-19} \pmod {99}$ $2n+1 \equiv -8 \pmod {99}$ yields $n=45$ as the smallest integer solution.

$2n+1 \equiv 8 \pmod {99}$ yields $n=53$ as the smallest integer solution.

$2n+1 \equiv -19 \pmod {99}$ yields $n=89$ as the smallest integer solution.

$2n+1 \equiv 19 \pmod {99}$ yields $n=9$ as the smallest integer solution. However, $n$ must be greater than $10$ .

The smallest positive integer solution greater than $10$ is $n=\boxed{045}$ .

Solution 3

$a_n=a_{n-1} + n \pmod{99}$ . Using the steps of the previous solution we get up to $n^2+n \equiv 90 \pmod{99}$ . This gives away the fact that $(n)(n+1) \equiv 0 \pmod{9} \implies n \equiv \{0, 8\} \pmod{9}$ so either $n$ or $n+1$ must be a multiple of 9.

Case 1 ( $n|9$ ): Say $n=9x$ and after simplification $x(9x+1) = 10 \pmod{90} \forall x \in \mathbb{Z}$ .

Case 2: ( $n+1|9$ ): Say $n=9a-1$ and after simplification $(9a-1)(a) = 10 \pmod{90} \forall a \in \mathbb{Z}$ .

As a result $a$ must be a divisor of $10$ and after doing some testing in both cases the smallest value that works is $x=5 \implies \boxed{045}$ .

~First

Solution 4 (not good, risky)

We just notice that $100 \equiv 1 \pmod{99}$ , so we are just trying to find $10 + 11 + 12 + \cdots + n$ modulo $99$ , or $\dfrac{n(n+1)}{2} - 45$ modulo $99$ . Also, the sum to $44$ is divisible by $99$ , and is the first one that is. Thus, if we sum to $45$ the $45$ is cut off and thus is just a sum to $44$ .

Without checking whether there are other sums congruent to $45 \pmod{99}$ , we can just write the answer to be $\boxed{045}$ .

@@ Line 43: / Line 43: @@
 ~First
+==Solution 4 (not good, risky)==
+We just notice that <math>100 \equiv 1 \pmod{99}</math>, so we are just trying to find <math>10 + 11 + 12 + \cdots + n</math> modulo <math>99</math>, or <math>\dfrac{n(n+1)}{2} - 45</math> modulo <math>99</math>. Also, the sum to <math>44</math> is divisible by <math>99</math>, and is the first one that is. Thus, if we sum to <math>45</math> the <math>45</math> is cut off and thus is just a sum to <math>44</math>.
+Without checking whether there are other sums congruent to <math>45 \pmod{99}</math>, we can just write the answer to be <math>\boxed{045}</math>.
 ==See also==
 {{AIME box|year=2017|n=I|num-b=8|num-a=10}}
 {{MAA Notice}}

2017 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search