2024 AMC 10A Problems/Problem 18
- 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 
positive integers 
with 
such that the base- integer 
is divisible by 
(where is in base ten). What is the sum of the digits of ?
Solution 1
, if even then 
. If odd then 
so 
. Now 
so 
but 
is too small so 
.
~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, is either even or odd. If is even, let 
.
\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 
for some integer 
, or 
.
What if is odd? Then let 
:
\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 
is odd. Thus, the other solution is 
for some integer , or 
.
We now simply must count the number of integers between 
and 
, inclusive, that are 
mod 
or mod 
. 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 
; this list is equivalent to 
, which comprises 
numbers. In the latter case, we have the numbers 
, which comprises 
numbers. There are 
numbers in total, so our answer is 
.
~Technodoggo
Solution 3
\begin{align*} 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)\\ \end{align*}
by the way I am sorry I deleted the original solution, how do I recover the original solution I wanted to add a new one actually
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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