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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # What is the remainder when 13^17 + 17^13 is divided by 10?  Question banks Downloads My Bookmarks Reviews Important topics
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Retired Moderator Joined: 07 Jun 2014
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GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
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Kudos [?]: 2971 , given: 394

What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]
Expert's post 00:00

Question Stats: 79% (01:03) correct 20% (02:13) wrong based on 24 sessions
What is the remainder when $$13^{17} + 17^{13}$$ is divided by 10?

[Reveal] Spoiler: OA
0

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Sandy
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Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]
1
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Remainder Property $$a^n$$ + $$b^n$$ is divisible by a+b if n is odd.
Here n=17 and 13 (odd). Now, a+b = 17+13 =30. Which is completely divisible by 10. Hence remainder is zero. Retired Moderator Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156 WE: Business Development (Energy and Utilities)
Followers: 173

Kudos [?]: 2971  , given: 394

Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]
1
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Expert's post
Explanation

The remainder when dividing an integer by 10 always equals the units digit. You can also ignore all but the units digits, so the question can be rephrased as: What is the units digit of $$3^{17} + 7^{13}$$?

The pattern for the units digits of 3 is [3, 9, 7, 1]. Every fourth term is the same. The 17th power is 1 past the end of the repeat: 17 – 16 = 1. Thus, $$3^{17}$$ must end in 3.

The pattern for the units digits of 7 is [7, 9, 3, 1]. Every fourth term is the same. The 13th power is 1 past the end of the repeat: 13 – 12 = 1. Thus, $$7^{13}$$ must end in 7. The sum of these units digits is 3 + 7 = 10. Thus, the units digit is 0.
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Sandy
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Intern Joined: 13 Jul 2018
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GRE 1: Q159 V151 GPA: 4
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Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]
Finding patterns is the key here.

17*1 = 1
17*17 = ..9
289*17 = ...3
...3*17 = ....1

Similarly, if we calculate the pattern for 13, we'll observe that it repeats in a similar way 13 does, after 4 multiplications.
So remainders will be (3+7)%10=0 Re: What is the remainder when 13^17 + 17^13 is divided by 10?   [#permalink] 17 Aug 2018, 16:18
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