1983 AIME Problems/Problem 6

Problem

Let $a_n$ equal $6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ .

Solution

First, we try to find a relationship between the numbers we're provided with and $49$ . We realize that $49=7^2$ and both $6$ and $8$ greater or less than 7 by $1$ .

Expressing the numbers in terms of $7$ , we get $(7-1)^{83}+(7+1)^{83}$ .

Applying the Binomial Theorem, half of our terms cancel out and we are left with $2(7^{83}+3403\cdot7^{81}+\cdots + 83\cdot7)$ . We realize that all of the terms in this big jumble of numbers are divisible by $49$ except the final term.

After some quick division, our answer is $35$ .

1983 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1983 AIME Problems/Problem 6

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