Question

use Euclid division lemma to show that any positive odd integers is of the form 6q+1, or 6q +3 or 6q+5, where q is some integers ​

Answer 1

r 0,1,2,4,5

by using eucluds division lemma

b<=r>b

a=bq

a=6q+o

a=6q+1

a=6q+2

a=6q+3

a=6q+4

a=6q+5

so 6q+0,6q+2,6q+4 is not an odd numbers

therefore the odd numbers is of the form 6q+1,6q+3,6q+5

Answer 2

We have to show that

♨️ Every positive odd integer is in the form of 6q + 1 , 6q + 3 and 6q + 5 ♨️

The general form of odd integers = 2n + 1

❄ 6q + 1

❄ 2(3q) + 1

❄ 2k + 1 ~ 2n + 1 [ 3q = k ]

:. 6q + 1 is also an odd integer

♨️ 6q + 3

♨️ 2 (3q) + 2 + 1

♨️ 2 (3q + 1) + 1

♨️ 2l + 1 ~ 2n + 1 [ 3q + 1 = l ]

:. 6q + 3 is also an odd integer

⭐ 6q + 5

⭐ 2 (3q) + 4 + 1

⭐ 2 (3q + 2) + 1

⭐ 2m + 1 ~ 2n + 1 [ 3q + 2 = m ]

:. 6q + 5 is also an odd integer

:. Every positive odd integer is in the form of 6q+1 , 6q+3 , 6q+5