2005 AMC 10B Problems/Problem 13

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 $\frac{5}{12} \cdot 2005$ = $\boxed{\textbf{(C) }835}$

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
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2005 AMC 10B Problems/Problem 13

Contents

Problem

Solution 1

Solution 2

See Also