Question

Q1) show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where Q is some integer.
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Answer 1

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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.

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Answered by : XxRainbowSparklexX

Answer 2

Step-by-step explanation:

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ANSWER

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Given ⤵

• Any positive odd integer is given to us.

To Show ⤵

• Any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where Q is some integer.

Solution ⤵

• Let 'a' be any integer
• And let the value of 'b' be 6.

Now , by Euclid's Algorithm , we have :

=> a = bq + r , where , 0 < r < b

=> a = 4q + r , where , 0 < r < 6 {Eq 1.}

So, we get the value of r as = 0 , 1 , 2 , 3 , 4 , 5.

• Now , putting the value of r in {Eq 1} one bye one , we obtain :

(1). a = 6q + 0 , { Even} , (Even)

(2). a = 6q + 1 , { Even + odd} , (Odd)

(3). a = 6q + 2 , { Even + Even} , (Even)

(4). a = 6q + 3 , { Even + odd} , (Odd)

(5). a = 6q + 4 , { Even + Even} , (Even)

(6). a = 6q + 5 , { Even + odd} , (Odd)

Now , in case 2, 4 and 6 number of above criteria we observe that they are only odd integer.

• And we also know that the product of any integer with even is even and with odd is odd.
• And addition of even with even is even. And sum of even with odd is Odd.
• And the sum of two odd is Even.

Hence , any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where Q is some integer.