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Show that any positive integer is form of 4q+1or4q+3, where q is some integer

Answer»

Question should be:

  • Show that any ODD positive INTEGER is form of 4q+1or4q+3, where q is some integer.

ANSWER:

  • All odd positive integer is in the form of 4q+1 , 4q+3 for some positive integer q.

GIVEN:

  • A positive integer q .

TO PROVE:

  • Any odd positive integer is form of 4q+1or4q+3, where q is some integer.

SOLUTION:

LET n be any positive integer which is divided by 4 we GET some quotient 'q' and remainder 'r'

=> n = 4q+r. .....(i)

Where r = 0, 1 ,2 ,3

Putting r = 0 in eq(i)

=> n = 4q (Which is even)

Putting r = 1

=> n = 4q+1. (Which is odd)

Putting r = 2

=> n = 2(2q+1) ..(which is even)

Putting r = 3

=> n = 4q+3 (which is odd)

Here any odd positive integer is in the form of 4q+1 , 4q+3 for some positive integer q.


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