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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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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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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).

Check Number Theory chapter of Math Book for more: math-number-theory-88376.html
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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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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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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.
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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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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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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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12 May 2014, 03:02
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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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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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12 May 2014, 03:14
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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?
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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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12 May 2014, 03:33

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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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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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.
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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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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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13 May 2014, 01:20
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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+)
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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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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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23 Jun 2015, 11:53
Hello from the GMAT Club BumpBot!

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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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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.
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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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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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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!
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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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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.
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Re: If a is a positive integer, and if the units digit of a^2 is   [#permalink] 14 Jul 2016, 10:15

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