Question

show that every integer is of the form 4q,4q+1,4q+2or 4q-1

Answer 1

let n be any positive integer, on dividing by 4 late q be the quotient and R be the remainder

then n is equal to 4 q + R where 0<r<4

therefore reminders may be 0 ,1, 2,and 3

case1: when R is equal to zero

=n = 4 q

=n is multiple of 2

so n is even

case 2:when r=1

=n=4q+1

=n=multiple of 2 + 1

=n=odd

case3:when r=2

=n=4q+2

=n=multiple of 2

n=even

case 4:when r=3

=n=4q+3

=n=(multiple of 2 +2)+1

=n=even +1

n=odd

case 5:when r=-1

n=4q-1

=n=(multiple of 2)-1

=n=even -1

n=odd

hence, every positive integers is of the form 4q, 4q+1 , 4q+2 or 4q-1