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# If a is an integer, what is the units digit of a^18?

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Manager
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If a is an integer, what is the units digit of a^18?  [#permalink]

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25 Nov 2014, 21:48
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Difficulty:

65% (hard)

Question Stats:

54% (01:43) correct 46% (01:24) wrong based on 142 sessions

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If a is an integer, what is the units digit of a^18?

(1) a^2 has a units digit of 9
(2) a^7 has a units digit of 3
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Re: If a is an integer, what is the units digit of a^18?  [#permalink]

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25 Nov 2014, 23:02
5
2
anceer wrote:
If a is an integer, what is the units digit of a^18?

1. a^2 has a units digit of 9
2. a^7 has a units digit of 3

The cyclicity of the last digit is at most 4.
e.g.
Last digit 2 - 2, 4, 8, 6 ... (cycle of 4)
Last digit 3 - 3, 9, 7, 1 ... (cycle of 4)
Last digit 4 - 4, 6, 4, 6 ... (cycle of 2)
Last digit 5 - 5, 5, 5, 5 ... (cycle of 1)
Last digit 6 - 6, 6, 6, 6 ... (cycle of 1)
Last digit 7 - 7, 9, 3, 1 ... (cycle of 4)
Last digit 8 - 8, 4, 2, 6 ... (cycle of 4)
Last digit 9 - 9, 1, 9, 1 ... (cycle of 2)
If last digit is 0 or 1, it remains 0 or 1 - cycle of 1

1. $$a^2$$ has a units digit of 9
In the cycle of 4, $$a^2$$ represents the second term. Last digit of a square is 9 in two cases.
Last digit 3 - 3, 9, 7, 1
Last digit 7 - 7, 9, 3, 1
So we don't know the last digit of a. But we are not asked the last digit of a. We are asked the last digit of $$a^{18}$$. It is also the second term in the cycle of 4. So it doesn't matter whether last digit of a is 3 or 7, the last digit of $$a^{18}$$ will be 9 in either case. Sufficient alone.

2. $$a^7$$ has a units digit of 3
In the cycle of 4, $$a^7$$ represents the third term. Last digit of the third term will be 3 in only one case - when the last digit of a is 7. So we can say that the last digit of $$a^{18}$$ must be 9. Sufficient alone.

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If a is an integer, what is the units digit of a^18?  [#permalink]

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Updated on: 28 Nov 2014, 12:40
2
(1) Sufficient. There are only two numbers -> 3 and 7 that can end in similar units digits when raised to a power of some number
3^2 = 9
3^3 = ..7 (multiplying just units digit 9 by 3)
3^4 = ..1 (multiplying just units digit 7 by 3)
3^5 = ..3 (multiplying just units digit 1 by 3)
3^6 = ..9
3^18 = ..9

7^2 = 49
7^3 = ..3 (multiplying just units digit 9 by 7)
7^4 = ..1 (multiplying just units digit 3 by 7)
7^5 = ..7
7^6 = ..9
7^9 = ..7
7^18 = ..9

(2) Sufficient. Only 7 has a units digit of 3 when raised to the power of 7.
7^1=7
7^2 = 49
7^3 = ..3 (multiplying just units digit 9 by 7)
7^4 = ...1 (multiplying just units digit by 7)
7^7 = ...3

3^1 = 3
3^2 = 9
3^3 = ..7
3^4 = ..1
3^7 = ..7

Originally posted by viktorija on 25 Nov 2014, 22:52.
Last edited by viktorija on 28 Nov 2014, 12:40, edited 1 time in total.
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Posts: 48037
Re: If a is an integer, what is the units digit of a^18?  [#permalink]

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26 Nov 2014, 05:54
anceer wrote:
If a is an integer, what is the units digit of a^18?

(1) a^2 has a units digit of 9
(2) a^7 has a units digit of 3

Similar questions to practice:
if-n-is-a-positive-integer-what-is-the-units-digit-of-n-154422.html
if-x-is-a-positive-integer-what-is-the-units-digit-of-x-94386.html

Also, check Units digits, exponents, remainders problems collection.

Hope it helps.
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Re: If a is an integer, what is the units digit of a^18?  [#permalink]

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28 Nov 2014, 01:10
1
(1) $$a^{18}=(a^2)^9=(*9)^9$$, so we can calculate the last digit. Sufficient.

(2) The units digit of $$a^7$$ could be 7 only if the units digit of $$a$$ is 7. To understand this, you need to check only odd last digits: 1,3,5,7,9. 1 and 5 doesn't change in any power. 9 in power can give only 1 or 9 as a units digit. $$3^7=3^4\times 3^3=*1\times*7=*7$$. So, the only one possible units digit is 7. Sufficicent

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Re: If a is an integer, what is the units digit of a^18?  [#permalink]

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12 Sep 2017, 23:19
Bunuel

Is there a way to do this quickly?
I got this question right but took me nearly 4 minutes.

I constructed a table of units digit for integers, similar to the explanations provided in this post.
Or is it one of those questions I should just guess and move on to save time?
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Posts: 48037
Re: If a is an integer, what is the units digit of a^18?  [#permalink]

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12 Sep 2017, 23:24
1
HappyQuakka wrote:
Bunuel

Is there a way to do this quickly?
I got this question right but took me nearly 4 minutes.

I constructed a table of units digit for integers, similar to the explanations provided in this post.
Or is it one of those questions I should just guess and move on to save time?

As you can see in the stats the average time of correct answer is 02:49, so after practice one should be able to solve it in less than 3 minutes.
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Re: If a is an integer, what is the units digit of a^18? &nbs [#permalink] 12 Sep 2017, 23:24
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