Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1991 AJHSME Problems/Problem 9"

(Created page with '==Problem== How many whole numbers from <math>1</math> through <math>46</math> are divisible by either <math>3</math> or <math>5</math> or both? <math>\text{(A)}\ 18 \qquad \te…')
 
Line 15: Line 15:
 
{{AJHSME box|year=1991|num-b=8|num-a=10}}
 
{{AJHSME box|year=1991|num-b=8|num-a=10}}
 
[[Category:Introductory Combinatorics Problems]]
 
[[Category:Introductory Combinatorics Problems]]
 +
{{MAA Notice}}

Revision as of 23:07, 4 July 2013

Problem

How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both?

$\text{(A)}\ 18 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 25 \qquad \text{(E)}\ 27$

Solution

There are $\left\lfloor \frac{46}{3}\right\rfloor =15$ numbers divisible by $3$, $\left\lfloor\frac{46}{5}\right\rfloor =9$ numbers divisible by $5$, so at first we have $15+9=24$ numbers that are divisible by $3$ or $5$, except we counted the multiples of $\text{LCM}(3,5)=15$ twice, once for $3$ and once for $5$.

There are $\left\lfloor \frac{46}{15}\right\rfloor =3$ numbers divisible by $15$, so there are $24-3=21$ numbers divisible by $3$ or $5$. $\rightarrow \boxed{\text{B}}$

See Also

1991 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1991_AJHSME_Problems/Problem_9&oldid=55604"