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What is the remainder when 7^442 is divided by 10?

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What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post Updated on: 22 Sep 2016, 22:14
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What is the remainder when 7^442 is divided by 10?

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

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Originally posted by MathRevolution on 22 Sep 2016, 17:29.
Last edited by Bunuel on 22 Sep 2016, 22:14, edited 1 time in total.
Renamed the topic and edited the question.
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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 22 Sep 2016, 22:15
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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 22 Sep 2016, 22:41
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Remainder of \(7^{442}\) is divided by 10 will be the last digit of \(7^{442}\)

\(7^1\) = \(7\)

\(7^2\)= \(49\)

\(7^3\)= \(xx3\)

\(7^4\)= \(xxx1\)

\(7^5\)=\(xxx7\)

\(7^6\)=\(xxx9\)

\(7^7\)=\(xxx3\)

\(7^8\) =\(xxx1\)
..............

if 7^n = xxxx1, if n is a multiple of 4.

\(7^{442}\) =\(7^{440}\)* \(7^2\) = xxx1 * 19 = xxxx9

\(\frac{7^{442}}{10} = \frac{(xxxx9)}{10} = xxxx + \frac{9}{10}\)--> remainder 9.

Ans E.
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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 22 Sep 2016, 18:46
MathRevolution wrote:
(integer) What is the remainder when 7442 is divided by 10?
A. 1 B. 3 C. 5 D. 7 E. 9


\(\frac{7442}{10} = 744 + \frac{2}{10}\)

Remainder should be 2. What I am missing here ? :roll: :?:
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What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 23 Sep 2016, 04:50
Find the units digit of 7^442 --> That would be the remainder

7 has 4 unit digit repetition ( as stated above by @0ld)
7^1 = 7
7^2 = 9
7^3 = 3
7^4/7^0 =1

Remainder of 442/4 is 2 --> 7^2 series = 9
E
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What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 24 Sep 2016, 22:14
==> From, ~71=~7, ~72=~9, ~73=~3, ~74=~1, the first digit has a cycle of ^4 in the order of 7-->9-->3-->1-->7. If so, from 442=4*110+2 you get the remainder of 2, and you get 7442=74*110+2=~72=~9. Hence, the answer is E.
Answer: E

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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 21 May 2019, 17:21
MathRevolution wrote:
What is the remainder when 7^442 is divided by 10?

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


Cyclicity of 7: 7, 9, 6, 1, 7...

442= 4K +2, so 7^2 will be left and hence last digit will be 9.
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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 22 May 2019, 00:12
How would you know to look for cyclical pattern here?

And can someone further explain?
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Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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New post 22 May 2019, 00:30
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cmccray26 wrote:
How would you know to look for cyclical pattern here?

And can someone further explain?


First Concept: any number when divided by 10 gives the last digit remainder. Check with numbers.

Second concept: In order to find the last digit we need to find a patter, in GMAT you can't calculate 7^442.

Now, how to find the cyclicity of numbers? try to understand the concepts: https://gmatclub.com/forum/cyclicity-92381.html#p711962

Next, For 7 you can check it yourself-
7^1, last digit 7
7^2, last digit 9
7^3, last digit 3
7^4, last digit 1
7^5, last digit 7...

So, for a number in the format 7^(4n+2) will have last digit 9. :)
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Re: What is the remainder when 7^442 is divided by 10?   [#permalink] 22 May 2019, 00:30
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