Question

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

Answer 1

Answer:

Step-by-step explanation:

Let a be any positive integer and b = 6.

By Euclid's algorithm ,a = bq + r.

a= 6q + r and q is some integer q ≥ 0 and r= 0,1,2,3,4 and 5.

Because 0 ≤ r < 6.

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

a = 6q +1 = 2 (3q) + 1 = 2m + 1 ( ∵where m= 3q is a positive integer )

a = 6q + 3 = 2 (3q + 1 ) + 1 = 2m + 1 ( ∵ where m = (3q+1) is a positive integer )

a = 6q + 5 = 2 (3q + 2 ) + 1 = 2m + 1 (∵ where m = (3q + 2 ) is a positive integer )

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

Hence,these expressions of numbers are odd numbers.

∴Any positive odd integer can be expressed in the form 6q + 1 or 6q + 3 or 6q + 5 is proved.