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Square of an Odd (Posted on 2002-10-06) Difficulty: 2 of 5
Take any odd number and square it. It will invariably be a multiple of 8 plus 1. So (odd)^2=8n+1 where n is an integer. Show why this is always so. Also show what the pattern for n is.

  Submitted by martyn    
Rating: 3.1333 (15 votes)
Solution: (Hide)
Z = an even number
(Z + 1) = odd
n = integer

(Z+1)^2 = z^2+2z+1

so we're trying to prove that
z^2 + 2z + 1 = 8n + 1

so
z^2 + 2z = 8n
since z is even, it must be 2w where w is an integer

(2w)^2 + 2(2w) = 8n
4w^2 + 4w = 8n
4*(w^2 + w) = 8n
4*[w*(w+1)] = 8n

8*[w*(w+1)/2] = 8n //notice that I turned the 4 into 8/2
[w*(w+1)]/2 = n

If you multiply two integers together, and at least one of them is even, the product will be even. Therefore, w*(w+1) is even, since one of them has to be even, and one of them has to be odd. So if [w*(w+1)] is even, then [w*(w+1)]/2 is an integer, equal to the n that we started with (8n+1)

It just so happens that [w*(w+1)]/2 = the sum of all the non-negative integers up to and including w (0 + 1 + 2 + 3 + ...... + w). So that is the pattern for n (0, 1, 3, 6, 10, 15).

1^2 = 8(0) + 1
3^2 = 8(1) + 1
5^2 = 8(3) + 1
7^2 = 8(6) + 1
9^2 = 8(10) + 1
11^2 = 8(15) + 1
.
.
.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionAnswerMath Man2013-06-10 16:51:54
SolutionPraneeth2007-08-01 13:11:24
SolutionPuzzle solution ( Continued)K Sengupta2007-04-20 12:23:11
SolutionPuzzle solutionK Sengupta2007-04-20 12:16:21
solAndre2005-09-07 22:03:39
Some Thoughtshmm..Amon2005-05-04 02:46:33
QuestionSimilar problemJonathan Fletcher2004-10-07 07:32:51
Some Thoughtsim smartBilly Bob2004-03-25 13:05:15
Solutioninduction does itAdy TZIDON2004-03-04 03:30:15
a simple solutionred_sox_fan_0320032004-03-03 17:56:18
SolutionPart 1,The formula for nTim Axoy2003-03-26 02:24:55
SolutionPart 1,The odd number 2m+1Tim Axoy2003-03-26 02:19:24
re: Alternate ApproachGamer2003-03-12 14:06:00
Not factorial!TomM2002-10-07 19:16:26
Square of an Odd NumberDouglas Johnson2002-10-07 15:47:22
Some ThoughtsAnswer to first questionDulanjana2002-10-07 14:46:57
SolutionAlternate ApproachTomM2002-10-06 17:52:23
re: no solution?TomM2002-10-06 17:33:36
no solution?martyn2002-10-06 17:19:53
SolutionHeh!TomM2002-10-06 13:25:45
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