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# What is the remainder when 13^17 + 17^13 is divided by 10?

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What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]  12 Aug 2018, 15:26
Expert's post
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Question Stats:

87% (00:51) correct 12% (00:00) wrong based on 8 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]  16 Aug 2018, 00:39
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.
GMAT Club Legend
Joined: 07 Jun 2014
Posts: 4749
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 93

Kudos [?]: 1652 [0], given: 396

Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]  17 Aug 2018, 16:05
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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Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]  17 Aug 2018, 16:18
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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