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

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

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New post 10 Feb 2011, 15:56
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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...
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If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 10 Feb 2011, 16:12
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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).

Answer: A.

Check Number Theory chapter of Math Book for more: math-number-theory-88376.html
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Re: Properties of Numbers  [#permalink]

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New post 10 Feb 2011, 16:33
Thanks Bunuel. It is more clear now.
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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
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Re: Properties of Numbers  [#permalink]

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New post 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).

Answer: A.

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.
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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..
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Re: Properties of Numbers  [#permalink]

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New post 08 Feb 2013, 13:36
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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.
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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!
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 12 May 2014, 03:14
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post Updated on: 03 Jun 2014, 19:23
Bunuel, Thanks for your reply. Yup, it does!

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.
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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.
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 12 May 2014, 03:40
MensaNumber wrote:
Bunuel, Thanks for your reply. Yup, it does!

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.
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New post 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!
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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.

Hence, the answer is 1.

Or using quadratic equation.

(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. :-D

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

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New post 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
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If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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! :o


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!
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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! :o


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.
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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)
ANSWER is A
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Re: If a is a positive integer, and if the units digit of a^2 is  [#permalink]

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New post 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.

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

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