Last visit was: 25 Apr 2026, 14:09 It is currently 25 Apr 2026, 14:09
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
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 25 Apr 2026
Posts: 109,831
Own Kudos:
811,266
 [25]
Given Kudos: 105,886
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: 109,831
Kudos: 811,266
 [25]
If A,BCD is a four-digit positive integer such that A,BCD is equal to 30 Aug 2015, 10:52
4
Kudos
Add Kudos
21
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

41% (02:34) correct 59% (02:39) wrong based on 202 sessions

History

Date
Time
Result
Not Attempted Yet
If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)

I. AD
II. CD
III. CA

(A) None
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III


Kudos for a correct solution.
ShowHide Answer
Official Answer
C
Most Helpful Reply
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 23 Apr 2026
Posts: 16,441
Own Kudos:
79,413
 [10]
Given Kudos: 485
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,441
Kudos: 79,413
 [10]
If A,BCD is a four-digit positive integer such that A,BCD is equal to 01 Sep 2015, 23:37
6
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
Bunuel
If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)

I. AD
II. CD
III. CA

(A) None
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III


Kudos for a correct solution.
Show more

Ideally, ABC * AB = ABCD should make you think of 1s and 0s. The digits needn't be unique so the first thing that comes to mind is 100*10 = 1000. But since CA and CD are given as two digit numbers, next you should think of 101*10 = 1010 or any such number. This satisfies all conditions so we can analyse using this example.

101 * 10 = 1010

AD = 10 - a factor of 1010
CD = 10 - a factor of 1010
CA = 11 - not a factor of 1010

Now the only thing left to figure is whether it is possible that C1 is a factor of 10C0 for some value of C.

10C0 = 1000 + C0 = 999 + C1
C1 is divisible by C1 but we don't know whether 999 is divisible by C1.
999 = 3^3*37
It has no two digit factor ending in 1 so it will not be divisible by C1.

Answer (C)



In case, 1s and 0s don't come to mind, here is how you could arrive at 10C0:

A bit of visualisation will help:

........A B C
........x A B
__________________
........M N P
.....Q R S x
__________________
.....A B C D


Note the leftmost digits: A*A gives you Q and you get A in the final product.
1*1 = 1 (possible value of Q and A)
2*2 = 4 (even if there were a carry over from previous calculation of M + R, Q+1 = 5, not A(2)). Hence A = 2 is not possible.
3*3 = 9 (even if there were a carry over from previous calculation, Q+1 would be 10 and the final product would be a 5 digit number.)

A cannot be any larger value because then the product would be a 5 digit number. Hence A must be 1 and there must be NO carryover from the previous calculation.

........1 B C
........x 1 B
__________________
........B N P
.....1 B C x
__________________
.....1 B C D

Now note that B+B = B. This is only possible if B = 0 or B = 9 (with a carryover from previous addition). But 9 + 9 + 1 = 19 and that would give a carryover to the next addition which we don't want. So B must be 0

So you get 10C * 10 = 10C0

This is true for all values of C from 1 to 9. (C cannot be 0 because CD is a two digit number)
User avatar
VenoMfTw
User avatar
Joined: 14 Mar 2014
Last visit: 15 Aug 2019
Posts: 133
Own Kudos:
487
 [8]
Given Kudos: 124
GMAT 1: 710 Q50 V34
GMAT 1: 710 Q50 V34
Posts: 133
Kudos: 487
 [8]
If A,BCD is a four-digit positive integer such that A,BCD is equal to 31 Aug 2015, 03:20
6
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Bunuel
If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)

I. AD
II. CD
III. CA

(A) None
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III


Kudos for a correct solution.
Show more


IMO : C

AB=10*A+B
ABC=100*A+10*B+C
ABCD=1000*A+100*B+10*C+D
AB*ABC=1000A^2 +100AB+10AC+100AB+10B^2+BC
=ABCD=1000*A+100*B+10*C+D
SO,A^2=A SO A=1
100AB=100B
10AC=10C
SO D=100AB+10B^2+BC
This is possible only if B=0 SO,D=0
Hence A=1,B=0,D=0 C can be any digit.
From options AD=10 which is always a factor of ABCD since D=0
CD may or may not be a factor ( if C = 5 then yes, if 6 then no)

CA cannot be a factor of ABCD. (If c = 5 then 51 not a factor of 1050)

So only CA cant be the factor.CD may be factor.
General Discussion
User avatar
rudra316
User avatar
Joined: 05 Jun 2015
Last visit: 02 May 2017
Posts: 61
Own Kudos:
94
 [1]
Given Kudos: 58
Location: United States
Schools: INSEAD Jan '17 ISB '17 NUS '18 HKUST '18
WE:Engineering (Transportation)
Schools: INSEAD Jan '17 ISB '17 NUS '18 HKUST '18
Posts: 61
Kudos: 94
 [1]
If A,BCD is a four-digit positive integer such that A,BCD is equal to 01 Sep 2015, 19:46
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ABCD = ABC * AB
1000A + 100B + 10C + D = (100A + 10 B + C)* (10A + B)
= 1000 A^2 + 100 (2AB) + 10( B^2 +AC) + BC
=> 1000 (A^2 - A) + 100 (2AB - B) + 10 ( AC + B^2 - B) + (BC - D) = 0
Given, A >0 ans as B, C , D are single digit integers which means(BC- D) will be atmost two digit number, for LHS to be zero, as the first two co-efficients are positive they need to be zero ( A^2 >=A & 2AB > B for all values).
2AB = B => A = 1/2 or B = 0.. As A is integer , B= 0
A^2 - A = 0 => A = 1

As D is product of B & C => D is 0

The number is of the form 10C0
Any value of C would be satisfy

AD (10 ) is factor of 10C0
CD (C0) if C is 3, 30 is not factor of 1030
CA (C1) if C is 1, 11 is not factor of 1010
Hence only AD is factor.

option D
avatar
optimusprime07
avatar
Joined: 01 Oct 2016
Last visit: 23 Aug 2025
Posts: 3
Own Kudos:
1
Posts: 3
Kudos: 1
If A,BCD is a four-digit positive integer such that A,BCD is equal to 27 Jul 2017, 09:04
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasPrepKarishma
Bunuel
If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)

I. AD
II. CD
III. CA

(A) None
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III


Kudos for a correct solution.

Ideally, ABC * AB = ABCD should make you think of 1s and 0s. The digits needn't be unique so the first thing that comes to mind is 100*10 = 1000. But since CA and CD are given as two digit numbers, next you should think of 101*10 = 1010 or any such number. This satisfies all conditions so we can analyse using this example.

101 * 10 = 1010

AD = 10 - a factor of 1010
CD = 10 - a factor of 1010
CA = 11 - not a factor of 1010

Now the only thing left to figure is whether it is possible that C1 is a factor of 10C0 for some value of C.

10C0 = 1000 + C0 = 999 + C1
C1 is divisible by C1 but we don't know whether 999 is divisible by C1.
999 = 3^3*37
It has no two digit factor ending in 1 so it will not be divisible by C1.

Answer (C)



In case, 1s and 0s don't come to mind, here is how you could arrive at 10C0:

A bit of visualisation will help:

........A B C
........x A B
__________________
........M N P
.....Q R S x
__________________
.....A B C D


Note the leftmost digits: A*A gives you Q and you get A in the final product.
1*1 = 1 (possible value of Q and A)
2*2 = 4 (even if there were a carry over from previous calculation of M + R, Q+1 = 5, not A(2)). Hence A = 2 is not possible.
3*3 = 9 (even if there were a carry over from previous calculation, Q+1 would be 10 and the final product would be a 5 digit number.)

A cannot be any larger value because then the product would be a 5 digit number. Hence A must be 1 and there must be NO carryover from the previous calculation.

........1 B C
........x 1 B
__________________
........B N P
.....1 B C x
__________________
.....1 B C D

Now note that B+B = B. This is only possible if B = 0 or B = 9 (with a carryover from previous addition). But 9 + 9 + 1 = 19 and that would give a carryover to the next addition which we don't want. So B must be 0

So you get 10C * 10 = 10C0

This is true for all values of C from 1 to 9. (C cannot be 0 because CD is a two digit number)
Show more



if we put 107 * 10 = 1070 , in this case C is 7 and D is 0
then 1070 is not divisible by 70
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 23 Apr 2026
Posts: 16,441
Own Kudos:
79,413
Given Kudos: 485
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,441
Kudos: 79,413
If A,BCD is a four-digit positive integer such that A,BCD is equal to 27 Jul 2017, 22:11
Kudos
Add Kudos
Bookmarks
Bookmark this Post
optimusprime07

if we put 107 * 10 = 1070 , in this case C is 7 and D is 0
then 1070 is not divisible by 70
Show more

Yes, but there exists a case in which it is. In the case of 101*10 = 1010, C = 1 and D = 0.
1010 is divisible by 10.
But the question asks: "which of the following two-digit numbers cannot be a factor of A,BCD?"
To qualify, CD should not be a factor of ABCD for all values of CD.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,985
Own Kudos:
1,118
Posts: 38,985
Kudos: 1,118
If A,BCD is a four-digit positive integer such that A,BCD is equal to 12 Aug 2025, 03:08
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.
Moderators:
Bunuel
Math Expert
109831 posts
Krunaal
Tuck School Moderator
852 posts