Divisible by 3,4 but not 5

March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)

0 replies

jlacosta

Mar 2, 2025

0 replies

Binomial Summation

Saucepan_man02 0

2 hours ago

Given that: $l_n = \sum_{r=1}^{n-1} \frac{2r^2-r(n-2)+1}{(n-r) \binom{n}{r}}$ , find $l_{15}$ .

0 replies

Saucepan_man02

2 hours ago

0 replies

Inequality with integers and indices

Michael Niland 5

N 2 hours ago by ohiorizzler1434

Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$ , and for a positive integer $n\geq3$ , prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$ .

5 replies

Michael Niland

Yesterday at 6:52 AM

ohiorizzler1434

2 hours ago

Quadratic Equation Problems

Saucepan_man02 4

N 2 hours ago by Saucepan_man02

4 replies

Saucepan_man02

Mar 3, 2025

Saucepan_man02

2 hours ago

Fun combinatorics

MTA_2024 3

N 2 hours ago by Amkan2022

A chess master plays $no$ $less$ $than$ $1$ $game$ $per$ $day$ , but no more than $10$ $games$ $per$ $week$
Prove that, if he plays long enough, we can find a series of consecutive days in which the master played $exactly$ $23$ $games$ .

3 replies

MTA_2024

Mar 13, 2025

Amkan2022

2 hours ago

No more topics!

Divisible by 3,4 but not 5

Silverfalcon 4

N Dec 4, 2005 by nonie

How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?

$\textbf{(A)} \ 768 \qquad \textbf{(B)} \ 801 \qquad \textbf{(C)} \ 934 \qquad \textbf{(D)} \ 1067 \qquad \textbf{(E)} \ 1167$

4 replies

Silverfalcon

Dec 3, 2005

nonie

Dec 4, 2005

Divisible by 3,4 but not 5

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

Silverfalcon

5006 posts

Silverfalcon

#1 h Dec 3, 2005, 2:26 AM • 2 Y

Y by Adventure10, Mango247

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

dangvh

374 posts

dangvh

#2 h Dec 3, 2005, 2:36 AM • 2 Y

Y by Adventure10, Mango247

Click to reveal hidden text

By dividing $2001$ by $3$ and $4$ and taking the integer part of the quotient we get $667$ and $500$ respectively. By the principle of inclusion and exclusion, we have to subtract the quotients when $2001$ is divided by $12$ , $15$ , and $20$ . Then add the quotient when $2001$ is divided by $60$ . Thus we get $667+500-166-133-100+33=801$ or $\boxed{B}$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

236factorial

4555 posts

236factorial

#3 h Dec 3, 2005, 3:16 AM • 1 Y

Y by Adventure10

Click to reveal hidden text

There are 667 multiples of 3 and 500 multiples of 4 from 0 - 2001, or 1167 total. The multiples of both three and four (or multiples of 12) are counted twice, so 1167 must be reduced by the number of multiples of 12 from 0 - 2001 or 166 numbers, leaving 1001 numbers. The problem states that multiples of 5 must be excluded, so we must eliminate the numbers that are multiples of both 3 and 5 (or multiples of 15), and numbers that are multiples of both 4 and 5 (or multiples of 20). After subtracting, there are 1001 - 133 - 100 = 768 numbers. We also eliminated the numbers that are multiples of 3, 4, and 5 (or multiples of 60) twice. We must add the number of multiples of 60 from 0 - 2001. There are 33, and adding this to 768 gives us 801 numbers.

This post has been edited 1 time. Last edited by 236factorial, Dec 4, 2005, 3:39 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

JesusFreak197

1939 posts

JesusFreak197

#4 h Dec 3, 2005, 5:26 AM • 2 Y

Y by Adventure10, Mango247

Answer

We take multiples of $3$ and multiples of $4$ , and subtract the overlap, so we have $667+500-166=1001$ . However, we must remove multiples of $5$ . Every $60$ , we have $15$ , $20$ , $30$ , $40$ , $45$ , $60$ that overlap, so we remove $6$ numbers per $60$ . There are 33 $60$ 's before $2001$ , so we subtract $198$ , and finally the $15$ and $20$ of the next $60$ , so we have $1001-200=801\Rightarrow \boxed{B}$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

nonie

834 posts

nonie

#5 h Dec 4, 2005, 12:54 AM • 1 Y

Y by Adventure10

Silverfalcon wrote:

How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?

$\text{(A)} \ 768 \qquad \text{(B)} \ 801 \qquad \text{(C)} \ 934 \qquad \text{(D)} \ 1067 \qquad \text{(E)} \ 1167$

Click to reveal hidden text

Take the # of multiples of 4 and add the # of multiples of 3 subtract the # of multiples of 3 and 4 (12) and you get the 3 or 4 part to get the non 5's you subtract the # of multiples of 3 and 5 (15) and the multiples of 4 and 5 (20) and finally add the # of multiples of all 3 numbers (60) (because we got rid of them twice).

So then you get .....

500+667-166-133-100+33=801.
SO the answer is B 801
Sometimes using a venn diagram helps me when I do these kind of problems.

Z K Y

N Quick Reply