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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] New post 12 Aug 2018, 15:26
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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
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Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink] New post 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.
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Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink] New post 17 Aug 2018, 16:05
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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] New post 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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What is the remainder when 13^17 + 17^13 is divided by 10?

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