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Chapter 4: Problem 44

Show that if \(n\) is an integer then \(n^{2} \equiv 0\) or 1\((\bmod 4)\)

Short Answer

Expert verified
If \(n\) is an integer, \(n^2 \equiv 0 \) or 1 \((\bmod \ 4)\).

Step by step solution

01

- Expressing Possible Remainders

Any integer n, when divided by 4, can have one of the following remainders: 0, 1, 2, or 3. This can be written as: \( n \equiv 0, 1, 2, \text{ or } 3 \pmod{4} \).
02

- Squaring Each Case

Now, square each possible remainder and reduce it modulo 4:1. If \( n \equiv 0 \pmod{4} \), then \( n^2 \equiv 0^2 \equiv 0 \pmod{4} \).2. If \( n \equiv 1 \pmod{4} \), then \( n^2 \equiv 1^2 \equiv 1 \pmod{4} \).3. If \( n \equiv 2 \pmod{4} \), then \( n^2 \equiv 2^2 \equiv 4 \equiv 0 \pmod{4} \).4. If \( n \equiv 3 \pmod{4} \), then \( n^2 \equiv 3^2 \equiv 9 \equiv 1 \pmod{4} \).
03

- Drawing the Conclusion

From the calculations in Step 2, we can see that the possible remainders for \( n^2 \) when divided by 4 are either 0 or 1. Therefore, \( n^2 \equiv 0 \) or \( 1 \pmod{4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

remainders
In modular arithmetic, the remainder is the number left over after division. When you divide an integer \( n \) by another integer, called the modulus, you get a quotient and a remainder. If you divide by 4, the possible remainders are 0, 1, 2, or 3.
For example:
- 7 divided by 4 gives a quotient of 1 and a remainder of 3.
- 10 divided by 4 gives a quotient of 2 and a remainder of 2.
We write this as \( 7 \equiv 3 \pmod{4} \) and \( 10 \equiv 2 \pmod{4} \).These remainders tell us how far an integer is from a multiple of the modulus.
modulus
The modulus is the number by which we divide to find the remainder. It sets the 'base' in modular arithmetic. Suppose we have a modulus of 4. Any integer can be expressed as running through a cycle of remainders when divided by 4: 0, 1, 2, and 3.
This repeats infinitely for every set of 4 numbers. For instance:
- 5 divided by 4 leaves a remainder of 1, so 5 is congruent to 1 modulo 4 (\( 5 \equiv 1 \pmod{4} \)).
- Similarly, 9 divided by 4 leaves a remainder of 1, so 9 is also congruent to 1 modulo 4 (\( 9 \equiv 1 \pmod{4} \)).
Both can be written as running through the same cycle relative to the modulus 4.
integer squares
When we square an integer, it follows a pattern in modular arithmetic. Let's examine this for the modulus 4. We know any integer \( n \) can be written as one of 0, 1, 2, or 3 modulo 4.
When we square these:
- \( 0^2 = 0 \rightarrow\ 0 \pmod{4} \)
- \( 1^2 = 1 \rightarrow\ 1 \pmod{4} \)
- \( 2^2 = 4 \rightarrow\ 0 \pmod{4} \)
- \( 3^2 = 9 \rightarrow\ 1 \pmod{4} \)
So, no matter the integer, its square can only be congruent to 0 or 1 modulo 4. This pattern helps in solving various problems in modular arithmetic, such as proving properties of numbers.

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Most popular questions from this chapter

Show that if \(p\) is an odd prime and \(a\) is an integer not divisible by \(p\) , then the congruence \(x^{2} \equiv a(\bmod p)\) has either no solutions or exactly two incongruent solutions modulo \(p .\)

Determine how we can use the decimal expansion of an integer \(n\) to determine whether \(n\) is divisible by \(\begin{array}{llll}{\text { a) } 2} & {\text { b) } 5} & {\text { c) } 10}\end{array}\)

Devise an algorithm that, given the binary expansions of the integers \(a\) and \(b\) , determines whether \(a>b, a=b,\) of \(a

Show that a positive integer is divisible by 3 if and only if the sum of its decimal digits is divisible by \(3 .\)

Encrypt the message GRIZZLY BEARS using blocks of five letters and the transposition cipher based on the permutation of \(\\{1,2,3,4,5\\}\) with \(\sigma(1)=3, \sigma(2)=5\) , \(\sigma(3)=1, \sigma(4)=2,\) and \(\sigma(5)=4 .\) For this exercise, use the letter \(X\) as many times as necessary to fill out the final block of fewer then five letters.
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