Last visit was: 24 Apr 2026, 07:59 It is currently 24 Apr 2026, 07:59
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
605-655 (Medium)|   Algebra|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 24 Apr 2026
Posts: 109,814
Own Kudos:
811,011
 [7]
Given Kudos: 105,871
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,814
Kudos: 811,011
 [7]
Let A, M, and C be nonnegative integers such that A + M + C = 12. What 15 Mar 2019, 01:47
3
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

45% (medium)

Question Stats:

69% (01:46) correct 31% (02:30) wrong based on 177 sessions

History

Date
Time
Result
Not Attempted Yet
Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A*M*C + A*M + M*C + A*C ?


(A) 62
(B) 72
(C) 92
(D) 102
(E) 112
ShowHide Answer
Official Answer
E
avatar
thyagi
avatar
Joined: 19 Jan 2019
Last visit: 28 Dec 2020
Posts: 80
Own Kudos:
64
Given Kudos: 8
GMAT 1: 650 Q49 V30
GMAT 1: 650 Q49 V30
Posts: 80
Kudos: 64
Let A, M, and C be nonnegative integers such that A + M + C = 12. What 15 Mar 2019, 02:35
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I got 112 .... I'm pretty not sure whether it's right or wrong... A+M+C = 12..
A=M=C=4

A*M*C + A*M +M*C +A*C
64+16+16+16 = 112...

Posted from my mobile device
User avatar
Archit3110
User avatar
Major Poster
Joined: 18 Aug 2017
Last visit: 24 Apr 2026
Posts: 8,629
Own Kudos:
5,190
 [1]
Given Kudos: 243
Status:You learn more from failure than from success.
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1: 545 Q79 V79 DI73
GMAT Focus 2: 645 Q83 V82 DI81
GPA: 4
WE:Marketing (Energy)
Products:
GMAT Club Tests Forum Quiz
GMAT Focus 2: 645 Q83 V82 DI81
Posts: 8,629
Kudos: 5,190
 [1]
Let A, M, and C be nonnegative integers such that A + M + C = 12. What 16 Mar 2019, 03:25
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A*M*C + A*M + M*C + A*C ?


(A) 62
(B) 72
(C) 92
(D) 102
(E) 112
Show more

let all values be equal = 4
so
64+16+16+16; 112
IMO E
User avatar
Zeus_
User avatar
Joined: 03 Jan 2024
Last visit: 23 Apr 2026
Posts: 17
Own Kudos:
8
Given Kudos: 6
Products:
GMAT Club Tests Forum Quiz
Posts: 17
Kudos: 8
Let A, M, and C be nonnegative integers such that A + M + C = 12. What Updated on: 18 Dec 2025, 04:21
Originally posted by Zeus_ on 18 Dec 2025, 04:04.
Last edited by Bunuel on 18 Dec 2025, 04:21, edited 1 time in total.
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A*M*C + A*M + M*C + A*C ?


(A) 62
(B) 72
(C) 92
(D) 102
(E) 112
Show more

To maximise the equation, you need to maximise A*M*C part as it represents the highest multiplication.

Combinations of 12:

10 + 1 + 1 = 12 [10*1*1 = 10]
8 + 2 + 2 = 12 [ 8*2*2 = 32]
6 + 3 + 3 = 12 [ 6*3*3 = 54]
4 + 4 + 4 = 12 [4^3 = 64]

Therefore, 64 + 16 + 16 + 16 = 64 + 16(3) = 112
User avatar
luisdicampo
User avatar
Joined: 10 Feb 2025
Last visit: 19 Apr 2026
Posts: 480
Own Kudos:
74
Given Kudos: 328
Products:
My Reviews Forum Quiz
Posts: 480
Kudos: 74
Let A, M, and C be nonnegative integers such that A + M + C = 12. What 11 Mar 2026, 03:27
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Deconstructing the Question

We want to maximize

\(AMC+AM+MC+AC\)

subject to

\(A+M+C=12\)

with A, M, and C nonnegative integers.

A useful rewrite is

\((A+1)(M+1)(C+1)=AMC+AM+MC+AC+A+M+C+1\)

Since

\(A+M+C=12\)

the expression becomes

\(AMC+AM+MC+AC=(A+1)(M+1)(C+1)-13\)

So we only need to maximize

\((A+1)(M+1)(C+1)\)

Step-by-step

Let

\(x=A+1,\ y=M+1,\ z=C+1\)

Then

\(x+y+z=15\)

and we want to maximize

\(xyz\)

For three integers with fixed sum, the product is greatest when the numbers are as equal as possible.

So the best choice is

\(x=5,\ y=5,\ z=5\)

Then

\(xyz=5\cdot5\cdot5=125\)

Now go back to the original expression:

\(AMC+AM+MC+AC=xyz-13\)

So

\(125-13=112\)

This happens when

\(A=4,\ M=4,\ C=4\)

Checking:

\(4\cdot4\cdot4+4\cdot4+4\cdot4+4\cdot4=64+16+16+16=112\)

Answer: E
Moderators:
Bunuel
Math Expert
109814 posts
Krunaal
Tuck School Moderator
853 posts