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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] New post 10 Feb 2011, 14:56
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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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Re: Properties of Numbers [#permalink] New post 10 Feb 2011, 15:12
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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 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
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Re: Properties of Numbers [#permalink] New post 10 Feb 2011, 15:33
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] New post 19 Apr 2012, 04:40
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] New post 06 Feb 2013, 18:35
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] New post 08 Feb 2013, 08:40
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] New post 08 Feb 2013, 12:36
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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.
Re: Properties of Numbers   [#permalink] 08 Feb 2013, 12:36
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