Is 550 the Correct Remainder for the Sum of the Series in Modular Arithmetic?

  • Thread starter saadsarfraz
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In summary, the conversation discusses finding the remainder when 1+2+2^{2}+...+2^{219} is divided by 5. The solution involves finding the congruence classes modulo 5 and identifying the pattern of the series. The correct answer is 0. An alternative approach is to directly sum the series using geometric series.
  • #1
saadsarfraz
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Q- What is the remainder when 1+2+2[tex]^{2}[/tex]+...+2[tex]^{219}[/tex] is divided by 5.

Solution: 2[tex]^{0}[/tex]=1 mod5
2[tex]^{1}[/tex]=2 mod5
2[tex]^{2}[/tex]=4 mod5
2[tex]^{3}[/tex]=3 mod5
2[tex]^{4}[/tex]=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.
 
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  • #2
Can anyone see if this is correct.
 
  • #3
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?
 
  • #4
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?
 
  • #5
Oops, I made a silly mistake. You are correct that the pattern 1,2,4,3 is repeating, since [itex]2^{m+4} \equiv 2^m ~(mod 5)[/itex].

So you have found out that:

[tex]\sum_{n=0}^{219}2^n \equiv 550~(mod 5) [/tex]

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).
 
  • #6
so 550 mod 5 is 0, as that leaves no remainder right?
 
  • #7
Correct.

Have you thought about the more direct approach, by summing the series?
 
  • #8
Geometric series?

Doesn't that sum to [tex] 2^{220} - 1 [/tex]?
 

FAQ: Is 550 the Correct Remainder for the Sum of the Series in Modular Arithmetic?

What is modular arithmetic?

Modular arithmetic is a branch of mathematics that deals with integers and their remainders when divided by a fixed number, also known as a modulus.

How does modular arithmetic differ from regular arithmetic?

Modular arithmetic follows the same basic principles as regular arithmetic, but it operates within a finite set of numbers defined by the modulus. This means that instead of using the traditional operations of addition, subtraction, multiplication, and division, we use modular addition, subtraction, multiplication, and division.

What are some real-life applications of modular arithmetic?

Modular arithmetic is used in various fields such as computer science, cryptography, and engineering. It is used to generate unique identification numbers, create secure encryption algorithms, and in coding theory to detect and correct errors in data transmission.

What is the purpose of the modulus in modular arithmetic?

The modulus in modular arithmetic serves as a fixed number that defines the set of numbers we are working with. It helps us to reduce large numbers to a smaller range, making calculations more manageable and efficient.

Can modular arithmetic be used with negative numbers?

Yes, modular arithmetic can be used with negative numbers. The modulus is still a positive integer, but the remainder can be negative. For example, in modular arithmetic with a modulus of 5, -3 is equivalent to 2 because -3 divided by 5 has a remainder of 2.

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