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Another equivalent to factoring

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• Consider the two arithmetic sequences k d2 + a2 and k + a1 where d2,a2,and a1 are integers, and k is the variable that takes on sequential values 0,1,2,3, .
Message 1 of 1 , Jun 15, 2008
Consider the two arithmetic sequences

k d2 + a2
and
k + a1

where d2,a2,and a1 are integers, and

k is the variable that takes on sequential values 0,1,2,3, . . .

We ask,

When does the term

(k + a1) divide ( k d2 + a2)?

We answer as follows:

k d2 + a2 = L ( k + a1) = L k + L a1

L k + L a1 - k d2 = a2

L k + L a1 - k d2 - a1 d2 = a2 - a1 d2

(L -d2) (k + a1) = a2 - a1 d2

We factor a2 - a1 d2 = x y

Then

L = d2 + x
k = y - a1

Conversely, if we wish to factor z = a2 - a1 d2

and discover ( accidentally or by design ) that

(k + a1) divides (k d2 + a2) then we know that

(k + a1) will also divide z.

Kermit Rose

< kermit@... >
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