Question

Show that any positive even integer is of the form 8p , 8p+2 , 8p-4 and 8p+6 where p is some integer.

Answer 1

Answer:

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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