ExplanationThe 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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