## Modular Arithematic-see if my answer is correct

Q- What is the remainder when 1+2+2$$^{2}$$+...+2$$^{219}$$ is divided by 5.

Solution: 2$$^{0}$$=1 mod5
2$$^{1}$$=2 mod5
2$$^{2}$$=4 mod5
2$$^{3}$$=3 mod5
2$$^{4}$$=1 mod5

Now I take (1,2,4,3) to be a set numbers. Since the summation goes to 219, there are a total of 220/4 = 55 sets. So I add 1+2+4+3= 10 and 10*55 = 550 <- my answer.
 PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity
 Can anyone see if this is correct.
 Recognitions: Gold Member Science Advisor Staff Emeritus No, it has a couple of errors. First, how many residues did you compute (4 or 5)? Or how many congruence classes are there modulo 5? Second, can the remainder exceed the divisor?

## Modular Arithematic-see if my answer is correct

i computed 4 residues, the pattern 1,2,3,4 keeps on repeating. you are right about remainder exceeding the divisor. any ideas about how i should compute this?
 Recognitions: Gold Member Science Advisor Staff Emeritus Oops, I made a silly mistake. You are correct that the pattern 1,2,4,3 is repeating, since $2^{m+4} \equiv 2^m ~(mod 5)$. So you have found out that: $$\sum_{n=0}^{219}2^n \equiv 550~(mod 5)$$ What is the least residue of 550 (mod 5)? That is the required answer. Now alternatively, you should also be capable of identifying the kind of series that is given to you and evaluating it directly (before looking at congruences).
 so 550 mod 5 is 0, as that leaves no remainder right?
 Recognitions: Gold Member Science Advisor Staff Emeritus Correct. Have you thought about the more direct approach, by summing the series?
 Geometric series? Spoiler Doesn't that sum to $$2^{220} - 1$$?

 Similar discussions for: Modular Arithematic-see if my answer is correct Thread Forum Replies Introductory Physics Homework 3 Introductory Physics Homework 3 Introductory Physics Homework 2 Calculus & Beyond Homework 2 Introductory Physics Homework 6