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Difference between revisions of "2020 AMC 10A Problems/Problem 22"

(→‎Solution (Casework))
(→‎Solution (Casework))
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Note that <math>n = 1</math> doesn't work; to prove this, we just have to substitute <math>1</math> for <math>n</math> in the expression.
 
Note that <math>n = 1</math> doesn't work; to prove this, we just have to substitute <math>1</math> for <math>n</math> in the expression.
 
This gives us
 
This gives us
−
<math>\left\lfloor \dfrac{998}{1} \right\rfloor+\left\lfloor \dfrac{999}{1} \right\rfloor+\left\lfloor \dfrac{1000}{1}\right \rfloor</math> = \frac{998}1 + \frac{999}1 + \frac{1000}1 = 2997 = 999 \cdot 3<math> which is divisible by 3.
+
<math>\left\lfloor \dfrac{998}{1} \right\rfloor+\left\lfloor \dfrac{999}{1} \right\rfloor+\left\lfloor \dfrac{1000}{1}\right \rfloor = \frac{998}1 + \frac{999}1 + \frac{1000}1 = 2997 = 999 \cdot 3</math> which is divisible by 3.
−
Therefore, the case </math>n = 1<math> does not work.
+
Therefore, the case <math>n = 1</math> does not work.
  
  
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−
<b>Case 1:</b> </math>n<math> divides </math>998<math>
+
<b>Case 1:</b> <math>n</math> divides <math>998</math>
  
−
The first case  does not work, as the three terms in the expression must be </math>(a, a, a)<math>, as mentioned above, so the sum becomes </math>3a<math>, which is divisible by </math>3<math>.
+
The first case  does not work, as the three terms in the expression must be <math>(a, a, a)</math>, as mentioned above, so the sum becomes <math>3a</math>, which is divisible by <math>3</math>.
  
  
−
<b>Case 2:</b> </math>n<math> divides </math>999<math>
+
<b>Case 2:</b> <math>n</math> divides <math>999</math>
  
−
Because </math>n<math> divides </math>999<math>, the number of possibilities for </math>n<math> is the same as the number of factors of </math>999<math>, excluding </math>1<math>.
+
Because <math>n</math> divides <math>999</math>, the number of possibilities for <math>n</math> is the same as the number of factors of <math>999</math>, excluding <math>1</math>.
−
</math>999<math> = </math>3^3 \cdot 37^1<math>
+
<math>999</math> = <math>3^3 \cdot 37^1</math>
−
So, the total number of factors of </math>999<math> is </math>4 \cdot 2 = 8<math>.
+
So, the total number of factors of <math>999</math> is <math>4 \cdot 2 = 8</math>.
  
−
However, we have to subtract </math>1<math>, because the case </math>n = 1<math> doesn't work, as mentioned previously.
+
However, we have to subtract <math>1</math>, because the case <math>n = 1</math> doesn't work, as mentioned previously.
−
</math>8 - 1 = 7<math>
+
<math>8 - 1 = 7</math>
  
 
We now do the same for the third and last case.
 
We now do the same for the third and last case.
  
  
−
<b>Case 3:</b> </math>n<math> divides </math>1000<math>
+
<b>Case 3:</b> <math>n</math> divides <math>1000</math>
  
−
Because </math>n<math> divides </math>1000<math>, the number of possibilities for </math>n<math> is the same as the number of factors of </math>1000<math>, excluding </math>1<math>.
+
Because <math>n</math> divides <math>1000</math>, the number of possibilities for <math>n</math> is the same as the number of factors of <math>1000</math>, excluding <math>1</math>.
−
</math>1000<math> = </math>5^3 \cdot 2^3<math>
+
<math>1000</math> = <math>5^3 \cdot 2^3</math>
−
So, the total number of factors of </math>1000<math> is </math>4 \cdot 4 = 16<math>.
+
So, the total number of factors of <math>1000</math> is <math>4 \cdot 4 = 16</math>.
  
−
Again, we have to subtract </math>1<math>, for the reason stated in Case 2.
+
Again, we have to subtract <math>1</math>, for the reason stated in Case 2.
−
</math>16 - 1 = 15<math>
+
<math>16 - 1 = 15</math>
  
  
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Now that we have counted all of the cases, we add them.
 
Now that we have counted all of the cases, we add them.
  
−
</math>7 + 15 = 22<math>, so the answer is </math>\boxed{\textbf{(A)}22}$.
+
<math>7 + 15 = 22</math>, so the answer is <math>\boxed{\textbf{(A)}22}</math>.
  
  

Revision as of 21:25, 1 February 2020

Problem

For how many positive integers $n \le 1000$ is\[\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor\]not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

$\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$


Solution (Casework)

Expression: \[\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor\]

Solution:

Let $a = \left\lfloor \frac{998}n \right\rfloor$

Notice that for every integer $n \neq 1$,

$\bullet$ if $\frac{998}n$ is an integer, then the three terms in the expression above must be $(a, a, a)$,

$\bullet$ if $\frac{999}n$ is an integer, then the three terms in the expression above must be $(a, a + 1, a + 1)$, and

$\bullet$ if $\frac{1000}n$ is an integer, then the three terms in the expression above must be $(a, a, a + 1)$.

This is due to the fact that $998$, $999$, and $1000$ share no common factors collectively (other than 1).


Note that $n = 1$ doesn't work; to prove this, we just have to substitute $1$ for $n$ in the expression. This gives us $\left\lfloor \dfrac{998}{1} \right\rfloor+\left\lfloor \dfrac{999}{1} \right\rfloor+\left\lfloor \dfrac{1000}{1}\right \rfloor = \frac{998}1 + \frac{999}1 + \frac{1000}1 = 2997 = 999 \cdot 3$ which is divisible by 3. Therefore, the case $n = 1$ does not work.


Now, we test the three cases mentioned above.


Case 1: $n$ divides $998$

The first case does not work, as the three terms in the expression must be $(a, a, a)$, as mentioned above, so the sum becomes $3a$, which is divisible by $3$.


Case 2: $n$ divides $999$

Because $n$ divides $999$, the number of possibilities for $n$ is the same as the number of factors of $999$, excluding $1$. $999$ = $3^3 \cdot 37^1$ So, the total number of factors of $999$ is $4 \cdot 2 = 8$.

However, we have to subtract $1$, because the case $n = 1$ doesn't work, as mentioned previously. $8 - 1 = 7$

We now do the same for the third and last case.


Case 3: $n$ divides $1000$

Because $n$ divides $1000$, the number of possibilities for $n$ is the same as the number of factors of $1000$, excluding $1$. $1000$ = $5^3 \cdot 2^3$ So, the total number of factors of $1000$ is $4 \cdot 4 = 16$.

Again, we have to subtract $1$, for the reason stated in Case 2. $16 - 1 = 15$


Now that we have counted all of the cases, we add them.

$7 + 15 = 22$, so the answer is $\boxed{\textbf{(A)}22}$.


~dragonchomper

Video Solution

https://youtu.be/Ozp3k2464u4

~IceMatrix

See Also

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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

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