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

Revision as of 18:58, 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 1

Let $a = \left\lfloor \frac{998}n \right\rfloor$ . If the expression $\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$ is not divisible by $3$ , then the three terms in the expression must be $(a, a + 1, a + 1)$ , which would imply that $n$ is a divisor of $999$ but not $1000$ , or $(a, a, a + 1)$ , which would imply that $n$ is a divisor of $1000$ but not $999$ . $999 = 3^3 \cdot 37$ has $4 \cdot 2 = 8$ factors, and $1000 = 2^3 \cdot 5^3$ has $4 \cdot 4 = 16$ factors. However, $n = 1$ does not work because $1$ a divisor of both $999$ and $1000$ , and since $1$ is counted twice, the answer is $16 + 8 - 2 = \boxed{\textbf{(A) }22}$ .

Solution 2 (Casework)

Let $a = \left\lfloor \frac{998}n \right\rfloor$ . Notice that for every integer $n \neq 1$ , if $\frac{998}n$ is an integer, then the three terms in the expression must be $(a, a, a)$ , if $\frac{998}n$ is an integer, then the three terms in the expression must be $(a, a + 1, a + 1)$ , and if $\frac{1000}n$ is an integer, then the three terms in the expression must be $(a, a, a + 1)$ . This is due to the fact that $998$ , $999$ , and $1000$ share no factors other than 1. So, there are two cases:

Case 1: $n$ 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 we have counted the case where $n = 1$ when we already know that $n \neq 1$ .

$8 - 1 = 7$

We now do the same for the second case.

Case 2: $n$ divides $1000$

$1000$ = $5^3 \cdot 2^3$

So, the total number of factors of $1000$ is $4 \cdot 4 = 16$ .

Again, we have to subtract $1$ , for the reason mentioned above in Case !.

$16 - 1 = 15$

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

$7 + 15 = 22$ , so the answer is

Video Solution

https://youtu.be/Ozp3k2464u4

~IceMatrix

@@ Line 13: / Line 13: @@
-<b>Case 1:</b> <math>n</math> divides 999
+<b>Case 1:</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>.
@@ Line 20: / Line 20: @@
 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 we have counted the case where <math>n = 1</math> when we already know that <math>n \neq 1</math>.
+<math>8 - 1 = 7</math>
+We now do the same for the second case.
+<b>Case 2:</b> <math>n</math> divides <math>1000</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>.
+Again, we have to subtract <math>1</math>, for the reason mentioned above in Case !.
+<math>16 - 1 = 15</math>
+Now that we have counted all of the cases, we add them.
+<math>7 + 15 = 22</math>, so the answer is
 ==Video Solution==

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
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