Difference between revisions of "2024 AMC 10A Problems/Problem 18"

Revision as of 22:08, 8 November 2024

The following problem is from both the 2024 AMC 10A #18 and 2024 AMC 12A #11, so both problems redirect to this page.

Problem

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?

$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solution 1

$2b^3+2b+4\equiv 0\pmod{16}\implies b^3+b+2\equiv 0\pmod 8$ , if $b$ even then $b+2\equiv 0\pmod 8\implies b\equiv 6\pmod 8$ . If $b$ odd then $b^2\equiv 1\pmod 8\implies b^3+b+2\equiv 2b+2\pmod 8$ so $2b+2\equiv 0\pmod 8\implies b+1\equiv 0\pmod 4\implies b\equiv 3,7\pmod 8$ . Now $8\mid 2024$ so $\frac38\cdot 2024=759$ but $3$ is too small so $759 - 1 = 758\implies\boxed{20}$ . ~OronSH ~mathkiddus ~andliu766

Solution 2

\begin{align*} 2024_b\equiv0\pmod{16} \\ 2b^3+2b+4\equiv0\pmod{16} \\ b^3+b+2\equiv0\pmod8 \\ \end{align*}

Clearly, $b$ is either even or odd. If $b$ is even, let $b=2a$ .

\begin{align*} (2a)^3+2a+2\equiv0\pmod8 \\ 8a^3+2a+2\equiv0\pmod8 \\ 0+2a+2\equiv0\pmod8 \\ a+1\equiv0\pmod4 \\ a\equiv3\pmod4 \\ \end{align*}

Thus, one solution is $b=2(4x+3)=8x+6$ for some integer $x$ , or $b\equiv6\pmod8$ .

What if $b$ is odd? Then let $b=2a+1$ :

\begin{align*} (2a+1)^3+2a+1+2\equiv0\pmod8 \\ 8a^3+12a^2+6a+1+2a+1+2\equiv0\pmod8 \\ 8a^3+12a^2+8a+4\equiv0\pmod8 \\ 4a^2+4\equiv0\pmod8 \\ a^2\equiv1\pmod2 \\ \end{align*}

This simply states that $a$ is odd. Thus, the other solution is $b=2(2x+1)+1=4x+3$ for some integer $x$ , or $b\equiv3\pmod4$ .

We now simply must count the number of integers between $5$ and $2024$ , inclusive, that are $6$ mod $8$ or $3$ mod $4$ . Note that the former case comprises even numbers only while the latter is only odd; thus, there is no overlap and we can safely count the number of each and add them.

In the former case, we have the numbers $6,14,22,30,\dots,2022$ ; this list is equivalent to $8,16,24,32,\dots,2024\cong1,23,4,\dots,253$ , which comprises $253$ numbers. In the latter case, we have the numbers $7,11,15,19,\dots,2023\cong4,8,12,16,\dots,2020\cong1,2,3,4,\dots,505$ , which comprises $505$ numbers. There are $758$ numbers in total, so our answer is $7+5+8=\boxed{\textbf{(D) 20}}$ .

~Technodoggo

Solution 4

$2024_b=2\ast\ b^3+2\ast\ b+4$ $2024_(b+8)=2\ast(b+8)^3+2\ast(b+8)+4$ ${2024}_{\left(b+8\right)}-{2024}_b=2*\left(8\right)*\left(b^2+8b+64\right)+2*8\ =16*\left(b^2+8b+64\right)+16$

$2024_(b+8)-2024_b\equiv0\ (mod\ 16)$ $2024_(b+8)\ \ \equiv2024_b\ \ (mod\ 16)$ $2024_0\equiv4\ (mod\ 16)$ $2024_1\equiv8\ (mod\ 16)$ $2024_2\equiv6\ (mod\ 16)$ $2024_3\equiv0(mod\ 16)$ $2024_4\equiv12(mod\ 16)$ $2024_5\equiv8(mod\ 16)$ $2024_6\equiv0(mod\ 16)$ $2024_7\equiv0(mod\ 16)$

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 48: / Line 48: @@
 ~Technodoggo
-==Solution 3==
+==Solution 4==
-Note that <math>2024_b=2b^3+2b+4</math> is to be divisible by <math>16</math>, which means that <math>b^3+b+2</math> is divisible by <math>8</math>.
-If <math>b=0</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
+<math>2024_b=2\ast\ b^3+2\ast\ b+4\ </math>
+<math>2024_(b+8)=2\ast(b+8)^3+2\ast(b+8)+4</math>
+<math>{2024}_{\left(b+8\right)}-{2024}_b=2*\left(8\right)*\left(b^2+8b+64\right)+2*8\ =16*\left(b^2+8b+64\right)+16</math>
-If <math>b=1</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
+<math>2024_(b+8)-2024_b\equiv0\ (mod\ 16)</math>
+<math>2024_(b+8)\ \ \equiv2024_b\ \ (mod\ 16)</math>
-If <math>b=2</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
+<math>2024_0\equiv4\ (mod\ 16)</math>
+<math>2024_1\equiv8\ (mod\ 16)</math>
-If <math>b=3</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
+<math>2024_2\equiv6\ (mod\ 16)</math>
+<math>2024_3\equiv0(mod\ 16)</math>
-If <math>b=4</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
+<math>2024_4\equiv12(mod\ 16)</math>
+<math>2024_5\equiv8(mod\ 16)</math>
-If <math>b=5</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
+<math>2024_6\equiv0(mod\ 16)</math>
+<math>2024_7\equiv0(mod\ 16)</math>
-If <math>b=6</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
-If <math>b=7</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
-Therefore, for every <math>8</math> values of <math>b</math>, <math>3</math> of them will make <math>b^3+b+2</math> divisible by <math>8</math>. Therefore, since <math>2024</math> is divisible by <math>8</math>, <math>\dfrac{3}{8}\cdot2024=759</math> values of <math>b</math>, but this includes <math>b=3</math>, which does not satisfy the given inequality. Therefore, the answer is <cmath>759-1=758\rightarrow7+5+8=\boxed{\text{(D) }20}</cmath> ~Tacos_are_yummy_1
 ==See also==

Page

Toolbox

Search