Question

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

Answer 1

Answer:  

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

Answer:

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