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

Latest revision as of 15:32, 14 August 2025

Problem

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

Solutions

Solution 1

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

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

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

Alternatively, we could have noted that $a^b\equiv a^{b\pmod{\phi{(n)}}}\pmod n$ . This way, we have $6^{83}\equiv 6^{83\pmod {42}}\equiv 6^{-1}\pmod {49}$ , and can finish the same way.

Solution 3 (cheap and quick)

As the value of $a$ is obviously $6^{83}+8^{83}$ we look for a pattern with others. With a bit of digging, we discover that $6^n+8^m$ where $m$ and $n$ are odd is equal to $\boxed{35}\:\text{mod}\:49$

-dragoon

-minor edits by flyingkinder123

Note: This isn't true?? Take $n=m=1,3,5,7$

Solution 4

$6^{83} + 8^{83} = (6+8)(6^{82}-6^{81}8+\ldots-8^{81}6+8^{82})$

Becuase $7|(6+8)$ , we only consider $6^{82}-6^{81}8+\ldots-8^{81}6+8^{82} \pmod{7}$

$6^{82}-6^{81}8+\ldots-8^{81}6+8^{82} \equiv (-1)^{82} - (-1)^{81}+ \ldots - (-1)^1 + 1 = 83 \equiv 6 \pmod{7}$

$6^{83} + 8^{83} \equiv 14 \cdot 6 \equiv \boxed{035} \pmod{49}$

Solution 5 last resort (bash)

Repeat the steps of taking modulo $49$ after reducing the exponents over and over again until you get a residue of $49,$ namely $35.$ This bashing takes a lot of time but it isn’t too bad. ~peelybonehead

Video Solution by OmegaLearn

https://youtu.be/-H4n-QplQew?t=792

~ pi_is_3.14

@@ Line 4: / Line 4: @@
 Let <math>a_n=6^{n}+8^{n}</math>. Determine the remainder upon dividing <math>a_ {83}</math> by <math>49</math>.
-== Solution ==
+== Solutions ==
 === Solution 1 ===
 Firstly, we try to find a relationship between the numbers we're provided with and <math>49</math>. We notice 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>.
@@ Line 33: / Line 33: @@
 Take <math>n=m=1,3,5,7</math>
-=== Solution 3===
+=== Solution 4===
 <math>6^{83} + 8^{83} = (6+8)(6^{82}-6^{81}8+\ldots-8^{81}6+8^{82})</math>
@@ Line 44: / Line 44: @@
-=== Solution 4 last resort (bash) ===
+=== Solution 5 last resort (bash) ===
 Repeat the steps of taking modulo <math>49</math> after reducing the exponents over and over again until you get a residue of <math>49,</math> namely <math>35.</math> This bashing takes a lot of time but it isn’t too bad. ~peelybonehead

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