Question

use division algorithm to show that any positive odd integer is the form of 6q+1 ,6q+3,6q+5

Answer 1

Answer:

let a be any positive integer and b=6. then by euclid algorithm, a=6q+r for q≥0 and r=0,1,2,3,4,5 because 0≤ r< 6

∴ a= 6q or 6q+1 or 6q+2 or 6q+3 or 6q+4 or 6q+5

also 6q+1=2(3q)+1 = 2p+1   where p=3q and p is any integer

6q+3= 6q+2+1 =2(3q+1)+1=2s+1 where s=3q+1 and s is any integer

6q+5=6q+4+1=2(3q+2)+1=2t+1 where t=3q+2 and t is any integer

∴6q+1,6q+3 and 6q+5 are not exactly divisible by 2

hence these expressions are of odd numbers

∴ any odd integer can be expressed in the form of 6q+1,6q+3 and 6q+5