Difference between revisions of "1983 AIME Problems/Problem 6"

Revision as of 09:56, 2 September 2013

Problem

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

Solution

Solution 1

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 $8$ are 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 these terms are divisible by $49$ except the final term.

After some quick division, our answer is $\boxed{035}$ .

Solution 2

Since $\phi(49) = 42$ (the Euler's totient function), by Euler's Totient Theorem, $a^{42} \equiv 1 \pmod{49}$ where $\text{gcd}(a,49) = 1$ . Thus $6^{83} + 8^{83} \equiv 6^{2(42)-1}+8^{2(42)-1}$ $\equiv 6^{-1} + 8^{-1} \equiv \frac{8+6}{48}$ $\equiv \frac{14}{-1}\equiv \boxed{035} \pmod{49}$ .

@@ Line 5: / Line 5: @@
 == Solution ==
 === Solution 1 ===
-First, we try to find a relationship between the numbers we're provided with and <math>49</math>. We realize that <math>49=7^2</math> and both <math>6</math> and <math>8</math> greater or less than <math>7</math> by <math>1</math>.
+First, we try to find a relationship between the numbers we're provided with and <math>49</math>. We realize that <math>49=7^2</math> and both <math>6</math> and <math>8</math> are greater or less than <math>7</math> by <math>1</math>.
 Expressing the numbers in terms of <math>7</math>, we get <math>(7-1)^{83}+(7+1)^{83}</math>.

1983 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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