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# Find the units digit of 3^{2018} - 2^{2018}.

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42
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Find the units digit of 3^{2018} - 2^{2018}.  [#permalink]

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21 Nov 2018, 03:39
00:00

Difficulty:

15% (low)

Question Stats:

71% (00:53) correct 29% (01:15) wrong based on 63 sessions

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[Math Revolution GMAT math practice question]

Find the units digit of $$3^{2018} - 2^{2018}.$$

$$A. 1$$
$$B. 3$$
$$C. 5$$
$$D. 7$$
$$E. 9$$

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" NUS School Moderator Joined: 18 Jul 2018 Posts: 1024 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Re: Find the units digit of 3^{2018} - 2^{2018}. [#permalink] ### Show Tags 21 Nov 2018, 03:45 3 has a cyclicity of 4. Then $$3^{2018}$$ is the same as the remainder when 2018 is divided by 4, which is 2. $$3^{2018} = 3^2$$ Similarly, 2 has a cyclicity of 4. then $$2^{2018} = 2^2$$ $$3^2-2^2$$ = 9-4 = 5 C is the answer. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8235 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Find the units digit of 3^{2018} - 2^{2018}. [#permalink] ### Show Tags 23 Nov 2018, 00:54 => The units digit is the remainder when $$3^{2018} - 2^{2018}$$ is divided by $$10$$. The remainders when powers of $$3$$ are divided by $$10$$ are $$3^1: 3,$$ $$3^2: 9,$$ $$3^3: 7,$$ $$3^4: 1,$$ $$3^5: 3,$$ So, the units digits of $$3^n$$ have period $$4$$: they form the cycle $$3 -> 9 -> 7 -> 1.$$ Thus, $$3^n$$ has the units digit of $$9$$ if $$n$$ has a remainder of $$2$$ when it is divided by $$4$$. The remainder when $$2018$$ is divided by $$4$$ is $$2$$, so the units digit of $$3^{2018}$$ is $$9$$. The remainders when powers of $$2$$ are divided by $$10$$ are $$2^1: 2,$$ $$2^2: 4,$$ $$2^3: 8,$$ $$2^4: 6,$$ $$2^5: 2,$$ So, the units digits of $$2^n$$ have period $$4$$: they form the cycle $$2 -> 4 -> 8 -> 6.$$ Thus, $$2^n$$ has the units digit of $$4$$ since $$n$$ has a remainder of $$2$$ when it is divided by $$4$$. The remainder when $$2018$$ is divided by $$4$$ is $$2$$, so the units digit of $$2^{2018}$$ is $$4$$. $$3^{2018} - 2^{2018}$$ has remainder $$9 – 4 = 5$$ when it is divided by $$10$$. Therefore, the answer is C. Answer: C _________________ 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$79 for 1 month Online Course"
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Re: Find the units digit of 3^{2018} - 2^{2018}.  [#permalink]

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23 Nov 2019, 04:28
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Re: Find the units digit of 3^{2018} - 2^{2018}.   [#permalink] 23 Nov 2019, 04:28
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