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.

(In reply to Square of an Odd Number by Douglas Johnson)

Although your basic math is right, your definition of m! is wrong, so the equations N² = 8m! +1 and n = m! are wrong.

m! = m(m - 1)(m -2)...(3)(2)(1)

m(m+1)/2 = T(m), the mth triangle number (1,3,6,10...)

If you substitute T(m) everywhere that you used m!, then your answer would be correct.

Incidently T(m) is related to m!. T(m - 1) = C(m,2), where C(a,b) is the combination (usually indicated by stacking a over b like a fraction without the division bar, and enclosing them in parentases) and is equal to a!/b!(a - b)!

Posted by TomM on 2002-10-07 19:16:26