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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42 GPA: 3.82
Find the units digit of 3^{2018} - 2^{2018}.  [#permalink]

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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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NUS School Moderator V
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]

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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

Math Revolution GMAT Instructor V
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]

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=>

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 $$10$$.

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Re: Find the units digit of 3^{2018} - 2^{2018}.  [#permalink]

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