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

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~Technodoggo
 
~Technodoggo
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==Solution 3==
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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>.
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If <math>b=0</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
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If <math>b=1</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
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If <math>b=2</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
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If <math>b=3</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
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If <math>b=4</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
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If <math>b=5</math>, then <math>b^3+b+2</math> is not divisible by <math>8</math>.
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If <math>b=6</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
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If <math>b=7</math>, then <math>b^3+b+2</math> is divisible by <math>8</math>.
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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==
 
==See also==

Revision as of 17:46, 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

$2b^3+2b+4\equiv 0\pmod{16}\implies b^3+b+2\equiv 0\pmod 8$, 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 $\tfrac38\cdot 2024=759$ but $3$ is too small so $758\implies\boxed{20}$. ~OronSH ~mathkiddus

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 3

Note that $2024_b=2b^3+2b+4$ is to be divisible by $16$, which means that $b^3+b+2$ is divisible by $8$.

If $b=0$, then $b^3+b+2$ is not divisible by $8$.

If $b=1$, then $b^3+b+2$ is not divisible by $8$.

If $b=2$, then $b^3+b+2$ is not divisible by $8$.

If $b=3$, then $b^3+b+2$ is divisible by $8$.

If $b=4$, then $b^3+b+2$ is not divisible by $8$.

If $b=5$, then $b^3+b+2$ is not divisible by $8$.

If $b=6$, then $b^3+b+2$ is divisible by $8$.

If $b=7$, then $b^3+b+2$ is divisible by $8$.

Therefore, for every $8$ values of $b$, $3$ of them will make $b^3+b+2$ divisible by $8$. Therefore, since $2024$ is divisible by $8$, $\dfrac{3}{8}\cdot2024=759$ values of $b$, but this includes $b=3$, which does not satisfy the given inequality. Therefore, the answer is \[759-1=758\rightarrow7+5+8=\boxed{\text{(D) }20}\] ~Tacos_are_yummy_1

See also

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

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