Show that any positive even integer is of the form 8p , 8p+2 , 8p-4 and 8p+6 where p is some integer.
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Euclid Division Lemma:
given two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition a = bq + r where 0 ≤ r ≤ b. ...
Given a=8p,8p+2,8p+4,8p+6
where b=8,
the possible remainders are 0,1,2,3,4,5,6,7.
a=8p+0
a=8p+1
a=8p+2
a=8p+3
a=8p+4
a=8p+5
a=8p+6
a=8p+7
where p is quotient....
so,the general form of an integer
a=8p+r
By the problem 'a' is even integer.....
therefore a cannot be 8p+1,8p+3,8p+5,8p+7
Hence,any odd integer is of the form 8p,8p+2,8p+4,8p+6........
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