
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
April 11th, 2018, 12:38 AM  #1 
Member Joined: Apr 2018 From: On Earth Posts: 34 Thanks: 0  Why is it that none of the square numbers is one less than a multiple of 3?
I've noticed that none of the square numbers is one less than a multiple of 3. I have a rough idea of why this is but I'm not completely sure. Please can someone clarify? Thanks. 
April 11th, 2018, 01:04 AM  #2  
Senior Member Joined: Aug 2012 Posts: 1,999 Thanks: 573  Quote:
Symbolically we write $n \equiv 1 \pmod 3$ or $n \equiv 2 \pmod 3$. Either way depending on what you need at the moment. Now what are the square numbers mod 3? * $0^2 \equiv 0 \pmod 3$. * $1^2 \equiv 1 \pmod 3$. * $2^2 \equiv 1 \pmod 3$. There are no other possibilities and none of these are 2. So one less than a multiple of 3 can never be a perfect square. Once you get the hang of this you can solve all kinds of similar problems. Last edited by Maschke; April 11th, 2018 at 01:15 AM.  
April 12th, 2018, 12:22 AM  #3  
Member Joined: Apr 2018 From: On Earth Posts: 34 Thanks: 0  Quote:
 
April 12th, 2018, 02:49 AM  #4  
Senior Member Joined: Oct 2009 Posts: 440 Thanks: 147  Quote:
In any case. For a number $n$, there are three possibilities: 1) $n$ is divisible by $3$, hence $n=3m$. Then $n^2=9m^2$ is also divisible by three, so can't be one less than a multiple of three. 2) $n$ is one less than a multiple of three, hence $n=3m1$. Then, $n^2 = (3m1)^2 = 9m^2  6m +1 = 3(3m^22m)+1$ This is one more than a multiple of three, so can't be one less than a multipple of three. 3) $n$ is one more than a multiple of three, hence $n=3m+1$. Then $n^2=(3m+1)^2= 9m^2+6m + 1 = 3(3m^2+2m)+1$. This is one more than a multiple of three, so again: it can't be one less than a multiple of three. So given any number, its square is never less than a multiple of three. Done.  

Tags 
multiple, numbers, square 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Square roots of numbers < 1  shunya  Algebra  12  March 30th, 2014 06:44 AM 
numbers in square puzzles  bash  Number Theory  2  March 26th, 2013 02:46 PM 
Added multiple numbers, one sum. Extraction?  aroratushar  Algebra  6  August 18th, 2010 03:57 AM 
Square Roots (2 digit numbers)  thatguy512  Algebra  1  December 1st, 2009 11:48 PM 
Smallest common multiple of numbers 1 through n  brunojo  Number Theory  2  November 19th, 2007 08:55 AM 