Difference between revisions of "2009 AMC 10B Problems/Problem 21"

Revision as of 00:13, 16 October 2024

Problem

What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8?

$\mathrm{(A)}\ 0\qquad \mathrm{(B)}\ 1\qquad \mathrm{(C)}\ 2\qquad \mathrm{(D)}\ 4\qquad \mathrm{(E)}\ 6$

Solution

Solution 1

The sum of any four consecutive powers of 3 is divisible by $3^0 + 3^1 + 3^2 +3^3 = 40$ and hence is divisible by 8. Therefore

$(3^2 + 3^3 + 3^4 + 3^5) + \cdots + (3^{2006} + 3^{2007} + 3^{2008} + 3^{2009})$

is divisible by 8. So the required remainder is $3^0 + 3^1 = \boxed {4}$ . The answer is $\mathrm{(D)}$ .

Solution 2

We have $3^2 = 9 \equiv 1 \pmod 8$ . Hence for any $k$ we have $3^{2k}\equiv 1^k = 1 \pmod 8$ , and then $3^{2k+1} = 3\cdot 3^{2k} \equiv 3\cdot 1 = 3 \pmod 8$ .

Therefore our sum gives the same remainder modulo $8$ as $1 + 3 + 1 + 3 + 1 + \cdots + 1 + 3$ . There are $2010$ terms in the sum, hence there are $2010/2 = 1005$ pairs of $1+3$ , and thus the sum is $1005 \cdot 4 = 4020 \equiv 20 \equiv \boxed{4} \pmod 8$ .

Solution 3

We have the formula $\frac{a(r^n-1)}{r-1}$ for the sum of a finite geometric sequence which we want to find the residue modulo 8. $\frac{1 \cdot (3^{2010}-1)}{2}$ $\frac{3^{2010}-1}{2} = \frac{9^{1005}-1}{2}$ $\frac{9^{1005}-1}{2} \equiv \frac{1^{1005}-1}{2} \equiv \frac{0}{2} \pmod 8$ Therefore, the numerator of the fraction is divisible by $8$ . However, when we divide the numerator by $2$ , we get a remainder of $4$ modulo $8$ , giving us $\mathrm{(D)}$ .

Note: you need to prove that $\frac{9^{1005}-1}{2}$ is not congruent to 0 mod 16 because if so, then the whole thing would be congruent to 0 mod 8, even after dividing by 2 ~ ilikepi12

Solution 4 (Patterns)

We can see that $/frac{3^0}{8}$ is has a remainder of 1. $/frac{3^1}{8}$ has a remainder of 4. $/frac{3^2}{8}$ also has a remainder of 4. $/frac{3^3}{8}$ has a remainder of 0.

If we keep on going, we can see that this pattern repeats every 4 powers of 3 - that is to say, the reminder cycles through $1$ , $4$ , $4$ , and $0$ .

Using simple math, we can see that if assume the pattern repeats every 4 numbers, $3^2009$ has a remainder of $\mathrm{(D)}$ .

Video Solution

https://youtu.be/V8VydUpAsS8

~savannahsolver

@@ Line 32: / Line 32: @@
 Note: you need to prove that <cmath>\frac{9^{1005}-1}{2}</cmath> is not congruent to 0 mod 16 because if so, then the whole thing would be congruent to 0 mod 8, even after dividing by 2 ~ ilikepi12
+== Solution 4 (Patterns) ==
+We can see that <math>/frac{3^0}{8}</math> is has a remainder of 1. <math>/frac{3^1}{8}</math> has a remainder of 4. <math>/frac{3^2}{8}</math> also has a remainder of 4. <math>/frac{3^3}{8}</math> has a remainder of 0.
+If we keep on going, we can see that this pattern repeats every 4 powers of 3 - that is to say, the reminder cycles through <math>1</math>, <math>4</math>, <math>4</math>, and <math>0</math>.
+Using simple math, we can see that if assume the pattern repeats every 4 numbers, <math>3^2009</math> has a remainder of <math>\mathrm{(D)}</math>.
 ==Video Solution==
 https://youtu.be/V8VydUpAsS8

2009 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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