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Difference between revisions of "2005 AMC 10B Problems/Problem 13"

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How many numbers between <math>1</math> and <math>2005</math> are integer multiples of <math>3</math> or <math>4</math> but not <math>12</math>?
 
How many numbers between <math>1</math> and <math>2005</math> are integer multiples of <math>3</math> or <math>4</math> but not <math>12</math>?
  
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<math>\mathrm{(A)} 501 \qquad \mathrm{(B)} 668 \qquad \mathrm{(C)} 835 \qquad \mathrm{(D)} 1002 \qquad \mathrm{(E)} 1169 </math>
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<math>\textbf{(A) } 501 \qquad \textbf{(B) } 668 \qquad \textbf{(C) } 835 \qquad \textbf{(D) } 1002 \qquad \textbf{(E) } 1169 </math>
  
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== Solution 1 ==
  
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== Solution ==
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To find the multiples of <math>3</math> or <math>4</math> but not <math>12</math>, you need to find the number of multiples of <math>3</math> and <math>4</math>, and then subtract twice the number of multiples of <math>12</math>, because you overcount and do not want to include them. The multiples of <math>3</math> are <math>\frac{2005}{3} = 668\text{ }R1.</math> The multiples of <math>4</math> are <math>\frac{2005}{4} = 501 \text{ }R1</math>. The multiples of <math>12</math> are <math>\frac{2005}{12} = 167\text{ }R1.</math> So, the answer is <math>668+501-167-167 = \boxed{\textbf{(C) } 835}</math>
  
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To find the multiples of 3 or 4 but not 12, you need to find the number of multiples of 3 and 4, and then subtract twice the number of multiples of 12, because you overcount and do not want to include them. The multiples of 3 are 2005/3 = 668 R1. The multiples of 4 are 2005/4 = 501 R1. The multiples of 12 are 2005/12 = 167 R1. So, the answer is 668+501-167-167 = <math>\boxed{\mathrm{(C)}\ 835}</math>
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== Solution 2 ==
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From <math>1</math>-<math>12</math>, the multiples of <math>3</math> or <math>4</math> but not <math>12</math> are <math>3, 4, 6, 8, </math>and <math>9</math>, a total of five numbers. Since <math>\frac{5}{12}</math> of positive integers are multiples of <math>3</math> or <math>4</math> but not <math>12</math> from <math>1</math>-<math>12</math>, the answer is approximately <math>\frac{5}{12} \cdot 2005</math> = <math>\boxed{\textbf{(C) }835}</math>
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== Solution 3==
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(Simpler version of solution 2)
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For every multiple of 12, there are a total of 7 multiples of 3 and 4 combined. However, we do not want to include the 2 multiples of 12, so we subtract 2 from 7, leaving us with 5.
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 +
We divide 2005 by 12 to determine how many such multiples exist:
 +
<cmath>\left\lfloor \frac{2005}{12} \right\rfloor = 167</cmath>
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Multiplying that by 5, we get:
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<cmath>167 \times 5 = 835</cmath>
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So our answer is:<cmath>\boxed{\textbf{(C) } 835}</cmath>
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 +
-LittleWavelet
  
 
== See Also ==
 
== See Also ==
 
{{AMC10 box|year=2005|ab=B|num-b=12|num-a=14}}
 
{{AMC10 box|year=2005|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:21, 6 August 2025

Problem

How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$?

$\textbf{(A) } 501 \qquad \textbf{(B) } 668 \qquad \textbf{(C) } 835 \qquad \textbf{(D) } 1002 \qquad \textbf{(E) } 1169$

Solution 1

To find the multiples of $3$ or $4$ but not $12$, you need to find the number of multiples of $3$ and $4$, and then subtract twice the number of multiples of $12$, because you overcount and do not want to include them. The multiples of $3$ are $\frac{2005}{3} = 668\text{ }R1.$ The multiples of $4$ are $\frac{2005}{4} = 501 \text{ }R1$. The multiples of $12$ are $\frac{2005}{12} = 167\text{ }R1.$ So, the answer is $668+501-167-167 = \boxed{\textbf{(C) } 835}$

Solution 2

From $1$-$12$, the multiples of $3$ or $4$ but not $12$ are $3, 4, 6, 8,$and $9$, a total of five numbers. Since $\frac{5}{12}$ of positive integers are multiples of $3$ or $4$ but not $12$ from $1$-$12$, the answer is approximately $\frac{5}{12} \cdot 2005$ = $\boxed{\textbf{(C) }835}$

Solution 3

(Simpler version of solution 2)

For every multiple of 12, there are a total of 7 multiples of 3 and 4 combined. However, we do not want to include the 2 multiples of 12, so we subtract 2 from 7, leaving us with 5.

We divide 2005 by 12 to determine how many such multiples exist: \[\left\lfloor \frac{2005}{12} \right\rfloor = 167\]

Multiplying that by 5, we get: \[167 \times 5 = 835\]

So our answer is:\[\boxed{\textbf{(C) } 835}\]

-LittleWavelet

See Also

2005 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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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