Last visit was: 11 Jul 2025, 02:28 It is currently 11 Jul 2025, 02:28
Close
GMAT Club Daily Prep
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.
Close
Request Expert Reply
Confirm Cancel
User avatar
kashishh
Joined: 02 Jun 2011
Last visit: 15 Oct 2019
Posts: 89
Own Kudos:
420
 [26]
Given Kudos: 11
Posts: 89
Kudos: 420
 [26]
2
Kudos
Add Kudos
24
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 11 July 2025
Posts: 102,634
Own Kudos:
740,299
 [8]
Given Kudos: 98,170
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,634
Kudos: 740,299
 [8]
3
Kudos
Add Kudos
5
Bookmarks
Bookmark this Post
General Discussion
User avatar
kashishh
Joined: 02 Jun 2011
Last visit: 15 Oct 2019
Posts: 89
Own Kudos:
Given Kudos: 11
Posts: 89
Kudos: 420
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 11 July 2025
Posts: 102,634
Own Kudos:
740,299
 [3]
Given Kudos: 98,170
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,634
Kudos: 740,299
 [3]
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
kashishh
Bunuel
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

Given: \(81=qa+1\) --> \(qa=80\) --> so, \(a\) is a factor of 80: 2, 4, 5, 8, 10, 16, 20, 40, or 80.

(1) The remainder when a is divided by 40 is 0 --> this basically tells us that \(a\) is a multiple of 40, so \(a\) could be either 40 or 80 (from the above). Not sufficient.

(2) The remainder when 40 is divided by a is 40 --> this basically means that \(a\) is greater than 40 (for example, the remainder upon division 40 by 50 is 40). Now, the only factot of 80 which is greater than 40 is 80 itself. So, \(a=80\). Sufficient.

Answer: B.

Hope it's clear.
Dear Bunuel,
i am not able to understand the st. 2 analysis. (ofcourse i understood what you have written.)
one doubt remains like: if it is said x/y leaves a remainder of 10, can i consider values of x/y as 10/20 and like that?

Yes, 10 divided by 20 yields the remainder of 10.

THEORY:
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).

For example positive integer n is divided by 25 yields the remainder of 13 can be expressed as: \(n=25q+13\). Now, the lowest value of \(q\) can be zero and in this case \(n=13\) --> 13 divided by 25 yields the remainder of 13. Generally when divisor (25 in our case) is more than dividend (13 in our case) then the reminder equals to the dividend. For example:
3 divided by 24 yields a reminder of 3 --> \(3=0*24+3\);
or:
5 divided by 6 yields a reminder of 5 --> \(5=0*6+5\),

Hope it's clear.
User avatar
ichauhan.gaurav
Joined: 14 May 2013
Last visit: 17 Jan 2014
Posts: 8
Own Kudos:
62
 [1]
Given Kudos: 3
Posts: 8
Kudos: 62
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

Given: \(81=qa+1\) --> \(qa=80\) --> so, \(a\) is a factor of 80: 2, 4, 5, 8, 10, 16, 20, 40, or 80.

(1) The remainder when a is divided by 40 is 0 --> this basically tells us that \(a\) is a multiple of 40, so \(a\) could be either 40 or 80 (from the above). Not sufficient.

(2) The remainder when 40 is divided by a is 40 --> this basically means that \(a\) is greater than 40 (for example, the remainder upon division 40 by 50 is 40). Now, the only factot of 80 which is greater than 40 is 80 itself. So, \(a=80\). Sufficient.

Answer: B.

Hope it's clear.


Thanks for the solution

I have one doubt ...
as per the 2nd statement when 40 is divided by a remainder is 40 i.e. 40 = aq +40 or aq =0 ?

can you please explain this also .. :(
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 11 July 2025
Posts: 102,634
Own Kudos:
Given Kudos: 98,170
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,634
Kudos: 740,299
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ichauhan.gaurav
Bunuel
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

Given: \(81=qa+1\) --> \(qa=80\) --> so, \(a\) is a factor of 80: 2, 4, 5, 8, 10, 16, 20, 40, or 80.

(1) The remainder when a is divided by 40 is 0 --> this basically tells us that \(a\) is a multiple of 40, so \(a\) could be either 40 or 80 (from the above). Not sufficient.

(2) The remainder when 40 is divided by a is 40 --> this basically means that \(a\) is greater than 40 (for example, the remainder upon division 40 by 50 is 40). Now, the only factot of 80 which is greater than 40 is 80 itself. So, \(a=80\). Sufficient.

Answer: B.

Hope it's clear.


Thanks for the solution

I have one doubt ...
as per the 2nd statement when 40 is divided by a remainder is 40 i.e. 40 = aq +40 or aq =0 ?

can you please explain this also .. :(

Yes, "the remainder when 40 is divided by a is 40" can be written as \(40=ap+40\) --> \(ap=0\). What is the point of confusion here?
User avatar
GyMrAT
Joined: 14 Dec 2017
Last visit: 03 Nov 2020
Posts: 413
Own Kudos:
495
 [2]
Given Kudos: 173
Location: India
Posts: 413
Kudos: 495
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
kashishh
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

(1) The remainder when a is divided by 40 is 0
(2) The remainder when 40 is divided by a is 40

Given a > 0 & 81 = ak + 1

a = ?

Statement 1:

The remainder when a is divided by 40 is 0

a = 40p = 40 or 80
to satisfy 81 = ak + 1

Statement 1 is Not Sufficient.

Statement 2:

The remainder when 40 is divided by a is 40

40 = aq + 40, hence a > 40 & has to satisfy 81 = ak + 1

Hence a = 80

Statement 2 is Sufficient.

Answer B.


Thanks,
GyM
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 13 May 2024
Posts: 6,755
Own Kudos:
34,056
 [1]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,755
Kudos: 34,056
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
kashishh
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

(1) The remainder when a is divided by 40 is 0
(2) The remainder when 40 is divided by a is 40

Target question: What is the value of a?

Given: 81 divided by a results in a remainder of 1
In other words, 81 is 1 greater than some multiple of a
This means that 80 is some multiple of a
Another way to say this is: a is a divisor of 80
The divisors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
So, the possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
ASIDE: I omitted 1 from the list, since it does not satisfy the condition of getting a remainder of 1 when 81 is divided by 1.

Statement 1: The remainder when a is divided by 40 is 0.
The possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
As you can see, there are two possible values of a satisfy statement 1.
Case a: a = 40. This works, because 40 divided by 40 leaves remainder 0. In this case, the answer to the target question is a = 40
Case b:a = 80. This works, because 80 divided by 40 leaves remainder 0. In this case, the answer to the target question is a = 80
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when 40 is divided by a is 40.
The possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
As you can see, ONLY ONE possible value of a satisfies statement 2.
When a = 80, we get a remainder of 40 when 40 is divided by 80
So, the answer to the target question is a = 80
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

RELATED VIDEO FROM OUR COURSE
User avatar
fskilnik
Joined: 12 Oct 2010
Last visit: 03 Jan 2025
Posts: 885
Own Kudos:
Given Kudos: 57
Status:GMATH founder
Expert
Expert reply
Posts: 885
Kudos: 1,693
Kudos
Add Kudos
Bookmarks
Bookmark this Post
kashishh
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

(1) The remainder when a is divided by 40 is 0
(2) The remainder when 40 is divided by a is 40
Very nice conceptual problem!

\(a \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\,\left( * \right)\)

\(81 = M \cdot a + 1\,\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,M \cdot a = 80\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,a \le 80\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{positive}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,80\,\,\,\,\left( {**} \right)\)

\(? = a\)


\(\left( 1 \right)\,\,a = 40J\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}} \left\{ \matrix{\\
\,{\rm{Take}}\,\,J = 1\,\,\,\,\left( {M = 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 40\,\,\,\,{\rm{viable}} \hfill \cr \\
\,{\rm{Take}}\,\,J = 2\,\,\,\,\left( {M = 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 80\,\,\,\,{\rm{viable}} \hfill \cr} \right.\)


\(\left( 2 \right)\,\,40 = N \cdot a + \underline {40} \,\,,\,\,N\,\,{\mathop{\rm int}} \,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\{ \matrix{\\
\,a\,\, > \,\,\underline {40} \hfill \cr \\
\,N = 0 \hfill \cr} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,a = 80\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
User avatar
OmerKor
Joined: 24 Jan 2024
Last visit: 01 Jul 2025
Posts: 138
Own Kudos:
Given Kudos: 149
Location: Israel
Posts: 138
Kudos: 149
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Bunuel,

Following up this Question b/c I tried solving it today.

Why does “a” can’t be multiple of 81 such as 162,243,324..
81/81x2 = 1/2. R=1
81/81x3 = 1/3. R=1

Posted from my mobile device
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 11 July 2025
Posts: 102,634
Own Kudos:
Given Kudos: 98,170
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,634
Kudos: 740,299
Kudos
Add Kudos
Bookmarks
Bookmark this Post
 
OmerKor
Hi Bunuel,

Following up this Question b/c I tried solving it today.

Why does “a” can’t be multiple of 81 such as 162,243,324..
81/81x2 = 1/2. R=1
81/81x3 = 1/3. R=1

Posted from my mobile device
­You cannot reduce like that when calculating a remainder. For example, 12 divided by 8 yields a remainder of 4. However, if you reduce each number by 4, you get 3 divided by 2, which results in a remainder of 1, not 4. Thus, the remainder was reduced by the same factor.

Hence, 81 divided by 162 results in a remainder of 81. Generally, when the divisor is greater than the dividend, the remainder is equal to the dividend.

6. Remainders



­
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 37,376
Own Kudos:
Posts: 37,376
Kudos: 1,010
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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.
Moderator:
Math Expert
102634 posts