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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
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What is the remainder when 7^442 is divided by 10?  [#permalink]

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Updated on: 22 Sep 2016, 22:14
00:00

Difficulty:

5% (low)

Question Stats:

79% (00:45) correct 21% (01:13) wrong based on 131 sessions

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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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"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" 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. Math Expert Joined: 02 Sep 2009 Posts: 60647 Re: What is the remainder when 7^442 is divided by 10? [#permalink] ### Show Tags 22 Sep 2016, 22:15 1 0ld wrote: 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 ? Edited the question. It's correct now. _________________ Manager Joined: 24 Aug 2016 Posts: 59 Location: India WE: Information Technology (Computer Software) Re: What is the remainder when 7^442 is divided by 10? [#permalink] ### Show Tags 22 Sep 2016, 22:41 1 1 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. _________________ "If we hit that bullseye, the rest of the dominos will fall like a house of cards. Checkmate." Manager Joined: 24 Aug 2016 Posts: 59 Location: India WE: Information Technology (Computer Software) Re: What is the remainder when 7^442 is divided by 10? [#permalink] ### Show Tags 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 ? _________________ "If we hit that bullseye, the rest of the dominos will fall like a house of cards. Checkmate." Manager Joined: 15 Apr 2016 Posts: 55 What is the remainder when 7^442 is divided by 10? [#permalink] ### Show Tags 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 _________________ Cheers, Shri ------------------------------- GMAT is not an Exam... it is a war .. Let's Conquer !!! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8454 GMAT 1: 760 Q51 V42 GPA: 3.82 What is the remainder when 7^442 is divided by 10? [#permalink] ### Show Tags 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 - Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. _________________ 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: What is the remainder when 7^442 is divided by 10?  [#permalink]

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

And can someone further explain?
ISB School Moderator
Joined: 08 Dec 2013
Posts: 643
Location: India
Concentration: Nonprofit, Sustainability
Schools: ISB
GMAT 1: 630 Q47 V30
WE: Operations (Non-Profit and Government)
Re: What is the remainder when 7^442 is divided by 10?  [#permalink]

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22 May 2019, 00:30
1
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.
Re: What is the remainder when 7^442 is divided by 10?   [#permalink] 22 May 2019, 00:30
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