Question

Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q is
some integer.
​

Answer 1

$\bold\pink{\fbox{\sf{Question}}}$

Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q is

some integer.

$\bold\red{\fbox{\sf{Solution}}}$

• According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.

• Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.

According to Euclid’s division lemma

a = bq + r

a = 6q + r………………….(1)

where, (0 ≤ r < 6)

So r can be either 0, 1, 2, 3, 4 and 5.

Case 1:

If r = 1, then equation (1) becomes

a = 6q + 1

The Above equation will be always as an odd integer.

Case 2:

If r = 3, then equation (1) becomes

a = 6q + 3

The Above equation will be always as an odd integer.

Case 3:

If r = 5, then equation (1) becomes

a = 6q + 5

The above equation will be always as an odd integer.

∴ Any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.

Hence proved.

Answer 2

Let

′

a

′

be any positive integer and b=6

Then by division algorithm

a=6q+r where r=0,1,2,3,4,5

so, a is of the form 6q or 6q+1 or 6q+2 or 6q+3 or

6q+3 or 6q+4 or 6q+5

Therefore If s is an odd integer

Then

′

a

′

is of the form 6q+1 or 6q+3 6q+5

Hence a positive odd integer is of the form 6q+1 or 6q+3 or 6q+5