Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders?

1. The range of the seven remainders is 6. 2. The seven numbers selected are consecutive integers

I've read the solutions both within the OG and within Manhattan GMAT's Official Guide Companion. I disagree with the answer they are presenting. If anyone can please explain the concept I'm missing I would greatly appreciate it!

FYI, when figuring out this problem, I chose the following seven consecutive integers 3, 4, 5, 6, 7, 8, 9. Based on my knowledge the sum of the remainders when divided by 7 = 3 (0 + 0 + 0 + 0 + 0 + 1 + 2 = 3). It's getting late so perhaps I'm missing something simple here, but choosing these numbers, from my perspective, negates the explanation/answer given.

Red part in your solution is not correct.

Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is non-negative integer and always less than divisor).

So according to above, when positive integer \(a\) is less than divisor \(d\) then remainder upon division \(a\) by \(d\) is always equals to \(a\), for example 5 divided by 10 yields reminder of 5. So when 1, 2, 3, 4, 5, or 6 is divided by 7 remainder is 1, 2, 3, 4, 5, and 6 respectively.

Or algebraically: 3 divided by 7 can be expressed as \(3=0*7+3\), so \(r=3\).

For complete solution of this problem above posts.

remainder = 1,2,3,4,5,6,0 => sum of remainders= 21

7, 13, 14, 21, 28, 35, 42 remainder = 0,6,0,0,0,0,0. = range 6 sum of remainders = 6

Statement 1 not sufficient. (2) The seven numbers selected are consecutive integers. Plugging numbers: 1,2,3,4,5,6,7 range - 6

remainder = 1,2,3,4,5,6,0 => sum of remainders= 21 10,11,12,13,14,15,16 => remainder = 3,4,5,6,0,1,2 => sum of remainders = 21. For any set of 7 consecutive integers divided by 7 the sum of remainders is always 21.

Seven different numbers are selected from the integers 1 to [#permalink]

Show Tags

28 Aug 2010, 07:07

Expert's post

2

This post was BOOKMARKED

udaymathapati wrote:

Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders? (1) The range of the seven remainders is 6. (2) The seven numbers selected are consecutive integers.

The trick here is to know that remainder is always non-negative integer less than divisor \(0\leq{r}<d\), so in our case \(0\leq{r}<7\).

So the remainder upon division of any integer by 7 can be: 0, 1, 2, 3, 4, 5, or 6 (7 values).

(1) The range of the seven remainders is 6 --> if we pick 6 different multiples of 7 (all remainders 0) and the 7th number 6 (remainder 6) then the range would be 6 and the sum also 6. But if we pick 7 consecutive integers then we'll have all possible remainders: 0, 1, 2, 3, 4, 5, and 6 and their sum will be 21. Not sufficient.

(2) The seven numbers selected are consecutive integers --> ANY 7 consecutive integers will give us all remainders possible: 0, 1, 2, 3, 4, 5, and 6. It does not matter what the starting integer will be: if it's say 11 then the remainder of 7 consecutive integers from 11 divided by 7 will be: 4, 5, 6, 0, 1, 2, and 3 and if starting number is say 14 then the remainder of 7 consecutive integers from 14 divided by 7 will be: 0, 1, 2, 3, 4, 5 and 6. So in any case sum=0+1+2+3+4+5+6=21. Sufficient.

Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders?

1. The range of the seven remainders is 6. 2. The seven numbers selected are consecutive integers

I've read the solutions both within the OG and within Manhattan GMAT's Official Guide Companion. I disagree with the answer they are presenting. If anyone can please explain the concept I'm missing I would greatly appreciate it!

FYI, when figuring out this problem, I chose the following seven consecutive integers 3, 4, 5, 6, 7, 8, 9. Based on my knowledge the sum of the remainders when divided by 7 = 3 (0 + 0 + 0 + 0 + 0 + 1 + 2 = 3). It's getting late so perhaps I'm missing something simple here, but choosing these numbers, from my perspective, negates the explanation/answer given. _________________

"Effort only fully releases its reward after a person refuses to quit." - Napoleon Hill

If my post helped you in any way please give KUDOS!

Thank you for the explanation, yes I believe my exhaustion was getting to me. I completely understand why I was calculating incorrect remainders. When we divide 3 by 7, 7 goes into 3, 0 times but there is three left over still, so r = 3. Very basic mistake on my part.

So taking my original set (referencing statement 2): {3, 4, 5, 6, 7, 8, 9} Remainders would be; 3, 4, 5, 6, 0, 1, 2 Therefore yeilding; 3 + 4 + 5 + 6 + 0 + 1 + 2 = 21

Thank you very much for the explanation and for the reference to other explanations. + 1 Kudos! _________________

"Effort only fully releases its reward after a person refuses to quit." - Napoleon Hill

If my post helped you in any way please give KUDOS!

Re: Seven different numbers are selected from the integers 1 to [#permalink]

Show Tags

23 Mar 2016, 17:42

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Final decisions are in: Berkeley: Denied with interview Tepper: Waitlisted with interview Rotman: Admitted with scholarship (withdrawn) Random French School: Admitted to MSc in Management with scholarship (...

Last year when I attended a session of Chicago’s Booth Live , I felt pretty out of place. I was surrounded by professionals from all over the world from major...

I may have spoken to over 50+ Said applicants over the course of my year, through various channels. I’ve been assigned as mentor to two incoming students. A...