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For a positive integer x, the units digits of (x+2)^2

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For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 17 Jul 2017, 02:00
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For a positive integer x, the units digits of \((x+2)^2\) and \((x+6)^2\) are 9. What is the units digit of \((x+3)^2\)?

A. 0
B. 2
C. 4
D. 6
E. 8

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For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 17 Jul 2017, 02:27
MathRevolution wrote:
For a positive integer x, the units digits of \((x+2)^2\) and \((x+6)^2\) are 9. What is the units digit of \((x+3)^2\)?

A. 0
B. 2
C. 4
D. 6
E. 8


(1) unit digit of sq. of any digit is 9 when the digit will be either 3 or 7

Thus substituting x=1 in \((x+2)^2\) and \((x+6)^2\)
we get unit digit=9

so unit digit of \((x+3)^2\) = (1+3)^2 =16 =6

Ans D
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Re: For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 17 Jul 2017, 02:29
MathRevolution wrote:
For a positive integer x, the units digits of \((x+2)^2\) and \((x+6)^2\) are 9. What is the units digit of \((x+3)^2\)?

A. 0
B. 2
C. 4
D. 6
E. 8


Digits 3 and 7 have the units digit of their square is 9...
So two cases..
1) x+2=3....X=1
X+2=7....X=5

2) x+6=_3....X=7
X+6=_7......X=1...

Common in two cases is unit's digit of x=1..
So units digit of x+3=1+3=4...
Units digit of its square =4^2=_6
D
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For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 17 Jul 2017, 07:44
1
Since we are talking about the square of terms having a unit digit 9,
lets evaluate which of numbers have squares that end with 9

3^2 = 9
7^2 = 49
13^2 = 169
17^2 = 289
23^2 = 529
27^2 = 729

Since the \((x+2)^2\) and \((x+6)^2\) have to be numbers ending 9,
x can be 1,11,21...

The units digit of \((x+3)^2\) for all these numbers will be always 6(Option D)
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Re: For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 19 Jul 2017, 01:59
==>It needs to satisfy \(x=~1\). Thus, you get \(~3^2=~9\) and \(~7^2=~9\). If you substitute \(X=~1\), you get \((x+3)^2=~4^2=~6\).

The answer is D.
Answer: D
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Re: For a positive integer x, the units digits of (x+2)^2  [#permalink]

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New post 20 Jul 2017, 16:30
MathRevolution wrote:
For a positive integer x, the units digits of \((x+2)^2\) and \((x+6)^2\) are 9. What is the units digit of \((x+3)^2\)?

A. 0
B. 2
C. 4
D. 6
E. 8


If the units digit of the square of a number is 9, that number’s units digit will be either 3 or 7. Thus, we know that the units digit of (x + 2) is 3 or 7 and the units digit of (x + 6) is 3 or 7. Note that (x + 2) and (x + 6) have a difference of 4; that is why we must have the units digit of (x + 2) as 3 and the units digit of (x + 6) as 7.

Since the units digit of (x + 3) is 3, we see that the units digit of x itself must be 1 (e.g., x = 1, 11, 21, etc.). Thus, the units digit of (x + 3)^2 is 6 (e.g., 4^2 = 16, 14^2 = 196, etc.).

Answer: D
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Re: For a positive integer x, the units digits of (x+2)^2   [#permalink] 20 Jul 2017, 16:30
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