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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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For a positive integer x, the units digits of (x+2)^2  [#permalink]

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17 Jul 2017, 02:00
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5% (low)

Question Stats:

90% (01:15) correct 10% (01:26) wrong based on 75 sessions

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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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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Intern Joined: 20 Jul 2015 Posts: 17 For a positive integer x, the units digits of (x+2)^2 [#permalink] ### Show Tags 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 Math Expert Joined: 02 Aug 2009 Posts: 7556 Re: For a positive integer x, the units digits of (x+2)^2 [#permalink] ### Show Tags 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 _________________ Senior PS Moderator Joined: 26 Feb 2016 Posts: 3386 Location: India GPA: 3.12 For a positive integer x, the units digits of (x+2)^2 [#permalink] ### Show Tags 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) _________________ You've got what it takes, but it will take everything you've got Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7224 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: For a positive integer x, the units digits of (x+2)^2 [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Re: For a positive integer x, the units digits of (x+2)^2  [#permalink]

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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.).

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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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