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Re: What is the remainder when 13^17 + 17^13 is divided by 10?
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17 Aug 2018, 16:05
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.