Is n^2 congruent to 0 or 1 (mod 3) for any integer n?

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Discussion Overview

The discussion centers on whether \( n^2 \) is congruent to 0 or 1 modulo 3 for any integer \( n \). It explores various approaches and reasoning related to this congruence, including cases based on the parity of \( n \) and examination of specific integer squares.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant asserts that \( n^2 \) is congruent to 0 or 1 modulo 3 for any integer \( n \).
  • Another participant suggests considering the parity of \( n \) (even or odd) to analyze its relationship with the congruence modulo 3.
  • A different approach involves examining the squares of integers from 1 to 9 modulo 3, proposing that this could be generalized to higher numbers due to the properties of powers of 10 modulo 3.
  • Additionally, a participant discusses the representation of numbers in base 3, noting that squares can only end in 0 or 1, which corresponds to the congruence conditions.
  • Another participant questions the forms of \( n \) when it is odd or even, inviting further exploration of these cases.

Areas of Agreement / Disagreement

Participants present multiple approaches and reasoning, but there is no consensus on a definitive conclusion regarding the congruence of \( n^2 \) modulo 3.

Contextual Notes

The discussion includes various methods and assumptions, such as the dependence on the parity of \( n \) and the specific properties of numbers in different bases, which may not be fully resolved.

phyguy321
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proof:
n^2 congruent 0 or 1 (mod3) for any integer n
 
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Try considering the two cases where n^2is either even or odd and how that relates to the congruence modulo 3.
 
Two other methods:

Consider just the squares of the integers from 1 to 9 modulo 3. Then you could generalize to higher numbers since powers of 10 are congruent to 1 (mod 3).

Or, consider a number in base 3. It can end in 0, 1 or 2. Thus a square in base 3 can only end in 0^2 = 0, 1^2 = 1, or 2^2 = 4 = 1 base 3. So a square in base 3 can only end in 0 or 1, which is equivalent to the square leaving a remainder of 0 or 1 upon division by 3.
 
Going along with what jeffreydk said, if n is an odd integer, what form does it have? what about if n is an even integer?
 

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