Question

show that every positive even integer is not in the form of 8p+1, 8p+3, 8p+5, 8p+7 by using (Edl) Euclid's division lemma a=bq+r.
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Answer 1

Answer:

8p + 1, 8p + 3, 8p + 5 and 8p + 7 is not an even positive integer as it forms 2n + 1. Inorder to be an even number it should be in the form of 2n

Step-by-step explanation:

Let's know about Euclid's Division Lemma.

It states that a = bq + r where a and b are any positive integers and 0 ≤r <b.

Let's know the form of Even and Odd number.

(i) Even Number

Even number is in the form of 2n where n is any integer.

Examples :

• 8 = 2 × 4 = 2(4) = 2n , n = 4
• 6 = 2 × 3 = 2(3) = 2n , n = 3

(ii) Odd Number

Odd number is in the form of (2n + 1) where n is any integer.

Examples :

• 7 = (2 × 3) + 1 = 2(3) + 1 = 2n + 1 , n = 3
• 19 = (2 × 18) + 1 = 2(18) + 1 = 2n + 1 , n = 18

Now coming to your question.

The question is asking to prove that 8p + 1, 8p + 3, 8p + 5 and 8p + 7 is not an even positive integer.

Let's take each expressions one by one

(i) 8p + 1

= (2 × 4p) + 1

= 2n + 1 where n = 4p

So 8p + 1 is not a even. It is an odd number as it is in the form of (2n + 1)

(ii) 8p + 3

= 2(4p + 1) + 1

= 2n + 1

= 2n + 1 where n = 4p + 1

So 8p + 3 is not a even. It is an odd number as it is in the form of (2n + 1)

(iii) 8p + 5

= 2(4p + 2) + 1

= 2n + 1

= 2n + 1 where n = 4p + 2

So 8p + 5 is not a even. It is an odd number as it is in the form of (2n + 1)

(iv) 8p + 7

= 2(4p + 3) + 1

= 2n + 1

= 2n + 1 where n = 4p + 3

So 8p + 7 is not a even. It is an odd number as it is in the form of (2n + 1)

Thanks !

Answer 2

Answer:

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