GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Sep 2019, 03:05

### 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

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

# If a is a positive integer, and if the units digit of a^2 is

Author Message
TAGS:

### Hide Tags

Intern
Joined: 12 Nov 2010
Posts: 19
If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

10 Feb 2011, 15:56
4
10
00:00

Difficulty:

(N/A)

Question Stats:

86% (01:15) correct 14% (01:40) wrong based on 611 sessions

### HideShow timer Statistics

If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A. 1
B. 3
C. 5
D. 6
C. 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...
Math Expert
Joined: 02 Sep 2009
Posts: 58106
If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

10 Feb 2011, 16:12
13
11
ChenggongMAS wrote:
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A) 1
B) 3
C) 5
D) 6
C) 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...

1234.567

1 - THOUSANDS
2 - HUNDREDS
3 - TENS
4 - UNITS
. - decimal point
5 - TENTHS
6 - HUNDREDTHS
7 - THOUSANDTHS

So thE units digit is the digit to the left of the decimal point or in integer it's the rightmost digit. For example: the units digit of 1.2 is 1 and the units digit of 13 is 3.

Back to the original question.
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

The units digit of a^2 is 9 --> the units digit of a itself is either 3 or 7 (3^2=9 and 7^2=49);
The units digit of (a+1)^2 is 4 --> the units digit of a+1 is either 2 or 8 (2^2=4 and 8^2=64), so the the units digit of a itself is either 2-1=1 or 8-1=7;

To satisfy both conditions the units digit of a must be 7. Now, a+2 will have the units digit equal to 7+2=9, thus the units digit of (a+2)^2, will be 1 (9^2=81).

Check Number Theory chapter of Math Book for more: math-number-theory-88376.html
_________________
##### General Discussion
Intern
Joined: 12 Nov 2010
Posts: 19

### Show Tags

10 Feb 2011, 16:33
Thanks Bunuel. It is more clear now.
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 176
Schools: Johnson '15
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

19 Apr 2012, 05:40
ChenggongMAS wrote:
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A. 1
B. 3
C. 5
D. 6
C. 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...

Substitution Method was followed to get the Answer A: 1

guess values: 3 satisfies 2nd condition, but not 3rd condition...
but 7 satisfied both conditions and hence the answer 1 was obatined due the sq of 9
_________________
Regards,
Harsha

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Satyameva Jayate - Truth alone triumphs
Senior Manager
Status: Gonna rock this time!!!
Joined: 22 Jul 2012
Posts: 428
Location: India
GMAT 1: 640 Q43 V34
GMAT 2: 630 Q47 V29
WE: Information Technology (Computer Software)

### Show Tags

06 Feb 2013, 19:35
Bunuel wrote:
ChenggongMAS wrote:
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A) 1
B) 3
C) 5
D) 6
C) 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...

1234.567

1 - THOUSANDS
2 - HUNDREDS
3 - TENS
4 - UNITS
. - decimal point
5 - TENTHS
6 - HUNDREDTHS
7 - THOUSANDTHS

So th units digit is the digit to the left of the decimal point or in integer it's the rightmost digit. For example: the units digit of 1.2 is 1 and the units digit of 13 is 3.

Back to the original question.
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

The units digit of a^2 is 9 --> the units digit of a itself is either 3 or 7 (3^2=9 and 7^2=49);
The units digit of (a+1)^2 is 4 --> the units digit of a+1 is either 2 or 8 (2^2=4 and 8^2=64), so the the units digit of a itself is either 2-1=1 or 8-1=7;

To satisfy both conditions the units digit of a must be 7. Now, a+2 will have the units digit equal to 7+2=9, thus the units digit of (a+2)^2, will be 1 (9^2=81).

Check Number Theory chapter of Math Book for more: math-number-theory-88376.html

I was wondering if there's any algebraic soln to this question.
_________________
hope is a good thing, maybe the best of things. And no good thing ever dies.

Who says you need a 700 ?Check this out : http://gmatclub.com/forum/who-says-you-need-a-149706.html#p1201595

My GMAT Journey : http://gmatclub.com/forum/end-of-my-gmat-journey-149328.html#p1197992
Intern
Joined: 15 Jan 2013
Posts: 25
Concentration: Finance, Operations
GPA: 4
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

08 Feb 2013, 09:40
ChenggongMAS wrote:
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A. 1
B. 3
C. 5
D. 6
C. 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...

For unit digit of a^2 to be 9...unit digit of a has to be 3 or 7...
Now for unit digit of (a+1)^2 to be 4..unit digit of a has to be 1 or 7....
From the above two conditions, unit value of a has to be 7, which will satisfy both the conditions...
Now id unit digit of a is 7, unit digit of (a+2)^2 hast to be 1..
Intern
Joined: 08 Oct 2012
Posts: 26

### Show Tags

08 Feb 2013, 13:36
2
Sachin9 wrote:
Bunuel wrote:

I was wondering if there's any algebraic soln to this question.

1. Units digit of a^2 is 9.

2. (a+1)^2 = a^2 + 2a + 1 .....UD of (a^2) + UD of (2a) + 1 = 4 ....9+1 +UD(2a)=4 .....10+UD(2a) = 4...therefore, a = 2 or 7

Based on 1 and 2, a can't be 2, so it has to be 7. We can calculate (a+2)^2

(a+2)^2 = a^2 + 4a + 4 = a^2 + 2(2a) +4 = UD(a^2)+2(UD of 2a) +4 ..this gives you units digit of 1...and thats the answer.
Retired Moderator
Joined: 29 Oct 2013
Posts: 257
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

12 May 2014, 03:05
Hi Bunuel, This one too is tagged as 'hard' in GMATPrep. While it is marked as sub 600 here. Thanks!
_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876
Math Expert
Joined: 02 Sep 2009
Posts: 58106
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

12 May 2014, 03:14
1
MensaNumber wrote:
Hi Bunuel, This one too is tagged as 'hard' in GMATPrep. While it is marked as sub 600 here. Thanks!

You are right but the difficulty level here is based on percentage of users who answered the question correctly/incorrectly: 89% of the users answered this question correctly. Hence the tag.

Does this make sense?
_________________
Retired Moderator
Joined: 29 Oct 2013
Posts: 257
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

Updated on: 03 Jun 2014, 19:23

However, I have not seen this logic hold true in every case. What are the percentage ranges for sub600, 600-700, +700 etc? This will help me point out incorrect tags if any so as to improve this forum.

And quite frankly I didnt find this question that easy. But cannot argue against the statistics unless those 101 users somehow were not representative of an average test taker. Moreover, I have heard GMAC also categorizes questions based on how many test takers got it right/wrong. With hundreds of thousands taking the gmat each year they are likely to have bigger data.

I am just trying to understand tagging here. Thanks for your understanding.
_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876

Originally posted by NoHalfMeasures on 12 May 2014, 03:33.
Last edited by NoHalfMeasures on 03 Jun 2014, 19:23, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 58106
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

12 May 2014, 03:40
MensaNumber wrote:

However, I have not seen this logic hold true in every case. What are the percentage ranges for sub600, 600-700, +700 etc? This will help me point out incorrect tags if any so as to improve this forum.

And quite frankly I didnt find this question that easy. But cannot argue against the statistics unless those 101 users somehow were not representative of an average test taker. Moreover, I have hard GMAC also categorizes questions based on how many test takers got it right/wrong. With hundreds of thousands taking the gmat each year they are likely to have bigger data.

I am just trying to understand tagging here. Thanks for your understanding.

Well, you can judge the difficulty level of a question based on the statistics and not on the tags. I agree that GMAC has larger data and their stats might be more representative. Having said that I must add that still the difficulty level is quite subjective issue.
_________________
Retired Moderator
Joined: 29 Oct 2013
Posts: 257
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

12 May 2014, 03:49
Yes difficulty is a subjective matter. Hence I think defining percentage ranges corresponding to sub600, 600-700 and +700 is a great way to bring in objectivity? Or do we have these ranges already? Thanks!
_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876
Math Expert
Joined: 02 Sep 2009
Posts: 58106
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

13 May 2014, 01:20
1
MensaNumber wrote:
Yes difficulty is a subjective matter. Hence I think defining percentage ranges corresponding to sub600, 600-700 and +700 is a great way to bring in objectivity? Or do we have these ranges already? Thanks!

% of incorrect answers - Difficulty
0 - 29 = low (sub-600)
30 - 69 = medium (600-700)
70 - 99 = hard (700+)
_________________
Retired Moderator
Joined: 29 Oct 2013
Posts: 257
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

13 May 2014, 01:44
Bunuel wrote:
% of incorrect answers - Difficulty
0 - 29 = low (sub-600)
30 - 69 = medium (600-700)
70 - 99 = hard (700+)

Great! Will follow those ranges. Thanks
_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876
Intern
Joined: 08 Feb 2014
Posts: 7
Location: United States
Concentration: Marketing, Operations
GMAT 1: 580 Q36 V34
WE: Information Technology (Computer Software)
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

07 Jul 2015, 21:28
kapsycumm wrote:
Sachin9 wrote:
Bunuel wrote:

I was wondering if there's any algebraic soln to this question.

1. Units digit of a^2 is 9.

2. (a+1)^2 = a^2 + 2a + 1 .....UD of (a^2) + UD of (2a) + 1 = 4 ....9+1 +UD(2a)=4 .....10+UD(2a) = 4...therefore, a = 2 or 7

Based on 1 and 2, a can't be 2, so it has to be 7. We can calculate (a+2)^2

(a+2)^2 = a^2 + 4a + 4 = a^2 + 2(2a) +4 = UD(a^2)+2(UD of 2a) +4 ..this gives you units digit of 1...and thats the answer.

We have been given a^2 = 9.
that means a = +/- 3.

We put +3 in (a+1)^2, it doesn't give us 4.
But if we put -3,

(-3+1)^2 = (-2)^2 = 4.

Similarly, in (a+2)^2 = (-3+2)^2 = (-1)^2 = 1.

(a+1)^2 = 4
a^2 + 2a(1) + 1^2 = 4
We know a^2 = 9.

2a = 4-9-1
2a = -6
a = -3.

Substitute, this in (a+2)^2. We get 1.

Let me know if it makes sense or my logic is flawed.
Manager
Joined: 24 Nov 2013
Posts: 57
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

20 Aug 2015, 02:51
Hanish Satija wrote:
We have been given a^2 = 9.
that means a = +/- 3.

Yes there is a flaw...a is a positive integer...it cannot be -3
Intern
Joined: 14 Jul 2016
Posts: 1
If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

14 Jul 2016, 09:55
Gentlemen,

Good afternoon.
It´s my first time in the Forum - I am glad to see such a nice resource!

Question: How can I be sure that if the units digit of (a^2 ) = 9 , for sure the units digit of "a" must be 3 or 7 ?

I have followed the answer by expanding the equations and adding the units digits, which I did too, but took quite a longer time.

My first thought when I saw the quation was this " units digit of "a" must e 7 or 9 " approach, however it just sounded in my mind like good a guess - how can I be sure that no other number squared from 0 to infinite will result in a number with 9 ,( or x, or y) in the units digit ? What theory am I missing, guys?

Thank you and luck to all!
Math Expert
Joined: 02 Sep 2009
Posts: 58106
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

14 Jul 2016, 10:15
itabra wrote:
Gentlemen,

Good afternoon.
It´s my first time in the Forum - I am glad to see such a nice resource!

Question: How can I be sure that if the units digit of (a^2 ) = 9 , for sure the units digit of "a" must be 3 or 7 ?

I have followed the answer by expanding the equations and adding the units digits, which I did too, but took quite a longer time.

My first thought when I saw the quation was this " units digit of "a" must e 7 or 9 " approach, however it just sounded in my mind like good a guess - how can I be sure that no other number squared from 0 to infinite will result in a number with 9 ,( or x, or y) in the units digit ? What theory am I missing, guys?

Thank you and luck to all!

If x is an integer to get the units digit of x^2 the only thing we need to know is the units digit of x itself. There are ten digits, so we can have only the following cases:

0^2 = 0
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81

As you can see only if an integer ends with 3 or 9 its square will have the units digit of 9.
_________________
Director
Joined: 04 Jun 2016
Posts: 558
GMAT 1: 750 Q49 V43
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

22 Jul 2016, 10:03
ChenggongMAS wrote:
If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

A. 1
B. 3
C. 5
D. 6
C. 14

I guess I am just not reading this properly. I don't understand what they mean by units digit...

$$a^2=9$$ ----> only possible if unit digit is 3 or 7 ($$3^2=9 ; 7^2=49$$)
$$(a+1)^2=4$$; means that a is 7 because (7+1) is 8 and $$8^2=64$$ (Unit digit is 4)
now a+2 = 7+2 =9
$$9^2= 81$$ (unit digit is 1)
_________________
Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.
Current Student
Joined: 12 Aug 2015
Posts: 2594
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

### Show Tags

23 Jan 2017, 17:53
Firstly -->
Units digit is a one digit number.
So option E is out.

Unit digit of x^2 is 9 => units digit of x can be 3 or 7
Units digit of (x+1)^2 is 4
Combing the above two statements =>
Hence the units digit of x must be 7.

Hence the units digit of (x+1)^2 will be => 1
Hence A.

_________________
Re: If a is a positive integer, and if the units digit of a^2 is   [#permalink] 23 Jan 2017, 17:53

Go to page    1   2    Next  [ 29 posts ]

Display posts from previous: Sort by