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# If a is a positive integer, and if the units digit of a^2 is 9 and the

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Intern
Joined: 09 Aug 2011
Posts: 24
If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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09 Aug 2011, 11:12
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Question Stats:

88% (00:50) correct 12% (01:25) wrong based on 75 sessions

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-is-a-positive-integer-and-if-the-units-digit-of-a-2-is-109036.html

[Reveal] Spoiler:
I've taken the same GMATPrep problem solving test twice. Despite the fact that I guessed the answer, I don't really understand the explanation behind this particular problem. What is the UNITS DIGIT!? Could anyone care to explain why the answer is what it is? Attached is the print screen shot of the problem.

Attachment:

question ps.jpg [ 24.37 KiB | Viewed 2175 times ]
[Reveal] Spoiler: OA
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Joined: 24 Jul 2011
Posts: 1358
GMAT 1: 780 Q51 V48
GRE 1: 1540 Q800 V740
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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09 Aug 2011, 11:35
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The units' digit is the first digit to the left of the decimal point.

For example, in 123.4567, the units digit is 3.
Similarly, in 234, the units digit is 4.

In the given question, if the units digit of a^2 is 9, it means the units digit of a can be 3 or 7 (because both 3^2 and 7^2 end in a units digit of 9). Now, the units digit of (a+1)^2 is given to be 4. This means the units digit of a must be 7 because the units digit of (7+1)^2 = 4, but the units digit of (3+1)^2 is not 4 (it is 6).

This means the units digit of a is 7, and therefore the units digit of a+2 is 9. So the units digit of (a+2)^2 = units digit of 9^2 = 1

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Intern
Joined: 09 Aug 2011
Posts: 24
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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09 Aug 2011, 11:48
thank you! this helped so much!
Manager
Joined: 26 Oct 2010
Posts: 81
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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09 Aug 2011, 14:29
Hi aks1985,
Consider the no 1234
Here
4 - units digit
3 - tens
2 - hundreds
1 - thousands
You might need that for other problems.
Director
Joined: 01 Feb 2011
Posts: 703
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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21 Aug 2011, 19:05
units digit of a^2 is 9

=> units digit of a is either 3 or 7

also given units digit of (a+1)^2 is 4. so units digit of a can be only be 7 ( 3 is ruled out)

=> units digit of (a+2)^2 is 1 (as (7+2)^2 = 81)

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Joined: 12 Aug 2015
Posts: 2343
GRE 1: 323 Q169 V154
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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24 Nov 2016, 03:25
Typical GMAT question.
Beautifully written.
unambiguous.

Here given=>
units digit of a^2 is 9
so the unit digit of a can either be 3 or be 7
unit digit of a+1 can be 4 or 8
so units digit of (a+1)^2 can be 6 or 4
but its given that units digit of (a+2)^2 is 4
hence units digit of a must be 7
so units digit of a+2 must be 9
and hence the units digit of (a+2)^2 will be 1

Hence A
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Math Expert
Joined: 02 Sep 2009
Posts: 43377
Re: If a is a positive integer, and if the units digit of a^2 is 9 and the [#permalink]

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24 Nov 2016, 03:29
aks1985 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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-is-a-positive-integer-and-if-the-units-digit-of-a-2-is-109036.html

[Reveal] Spoiler:
I've taken the same GMATPrep problem solving test twice. Despite the fact that I guessed the answer, I don't really understand the explanation behind this particular problem. What is the UNITS DIGIT!? Could anyone care to explain why the answer is what it is? Attached is the print screen shot of the problem.

Attachment:
question ps.jpg

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-is-a-positive-integer-and-if-the-units-digit-of-a-2-is-109036.html
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Re: If a is a positive integer, and if the units digit of a^2 is 9 and the   [#permalink] 24 Nov 2016, 03:29
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