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# What is the ones digit of 3^23 - 2^18?

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What is the ones digit of 3^23 - 2^18? [#permalink]  27 Jul 2017, 10:05
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Question Stats:

72% (01:26) correct 27% (01:53) wrong based on 11 sessions

What is the ones digit of $$3^{23} - 2^{18}$$ ?

[Reveal] Spoiler:
3

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Re: What is the ones digit of 3^23 - 2^18? [#permalink]  25 Sep 2017, 06:11
Let's use the fact that unit digits are recurrent in powers so that
$$3^1=3$$
$$3^2=9$$
$$3^3=..7$$
$$3^4=..1$$
$$3^5=..3$$
...

And
$$2^1=2$$
$$2^2=4$$
$$2^3=8$$
$$2^4=..6$$
$$2^5=..2$$
....

Thus, $$3^21=...7$$ and $$2^18=...4$$. Then, their difference will terminate with a 7-4=3!
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Re: What is the ones digit of 3^23 - 2^18? [#permalink]  25 Sep 2017, 06:30
Carcass wrote:
What is the ones digit of $$3^{23} - 2^{18}$$ ?

[Reveal] Spoiler:
3

Now we know

Ones digit of$$2 ^ {any power}$$ = 2,4,8,6 i.e it keeps repeating after every 4 cycle

Ones digit of $$3 ^ {any power}$$ = 3,9,7,1 i.e it keeps repeating after every 4 cycle

Now ones digit of $$3^{23}$$ = 7 (divide 23/4 , since it repeats after every 4 cycle and the remainder is 3, so we have to consider the third term i.e 7 (3,9,7,1))

Similarly ones digit of $$2^{18}$$ = 4 (divide 18/4 , since it repeats after every 4 cycle and the remainder is 2,
so we have to consider the second term i.e 4(2,4,8,6))

Now

the ones digit of $$3^{23} - 2^{18}$$ = 7-4 = 3
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Re: What is the ones digit of 3^23 - 2^18?   [#permalink] 25 Sep 2017, 06:30
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