Question

show that any positive odd integer is of the form 6 m + 1 or 6 m + 3 or 6 m + 5 Where M is some integer​

Answer 1

Let a be a positive integer which is divided by 6 and m and r are the quotient and remainder respectively; then by Euclid's division lemma,

$a = 6m + r \: \: \: \: where \: \: \: 0 \leqslant r < 6 \\ or \: \: a = 6m \: \: or \: \: a = 6m + 1 \: \: or \: \: a = 6m + 2 \\ or \: \: a = 6m + 3 \: \: or \: \: a = 6m + 4 \: \: or \: \: a = 6m + 5$

But; 6m, 6m + 2 and 6m + 4 are the even values of a.

Hence, every odd integer a can be represented in the form (6m+1), (6m+3) and (6m+5)

HOPE THIS COULD HELP!!!

Answer 2

Step-by-step explanation:

To show: any positive odd int. is of the form 6m+1 , 6m+3 or 6m+5 .

Let 'a' be any positive odd int. ,

Such that on being divided by the digit 6 ,

'm' is the quotient and 'r' is the remainder .

Therefore, By Euclid's Division Lemma ,

a = 6m + r , where 0> or = r < 6

Therefore, the possible values of 'r' are ,

r=0,

r=1,

r=2,

r=3,

r=4,

r=5

Therefore, all possible values of 'a' are ,

a=6m+0=6m,

a=6m+1,

a=6m+2,

a=6m+3,

a=6m+4,

a=6m+5

But ,

a=6m,

a=6m+2,

a=6m+4

are all positive ~even~ values of 'a' .

Therefore , 'a' being a positive odd int. is of the form ,

6m+1,

6m+3,

or 6m+5 .

Therefore , it is proved that any positive odd int. is of the form 6m+1 , 6m+3 or 6m+5 .