Question

show that every positive even integers is of the form 4q, 4q+2 and that every positive odd integers is of the form of 4q+1 and 4q+3 ,where q is some integers​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

1) Show that any positive even integer is of the form 4q or 4q+2.

Let a and b be any positive even integer. (a > b)

Then by Euclid's division lemma.

a = bq + r

Take b = 4 (0 ≤ r < b)

a = 4q + r

Where r = 0, 1, 2, 3

When r = 0

a = 4q + (0)

a = 4q = 2(2q) is an even number

When r = 2

a = 4q + (2)

Take 2 as common

a = 2(2q + 1) which is also an even number

So, we can say that any positive even integer is in the form 4q or 4q + 2 where q is any integer.

2) Show that any positive odd integer is of the form 4q+1 or 4q+3 where q is any integer.

Let a and b be any odd integer. (a > b)

Then by Euclid's division lemma.

a = bq + r

Take b = 4 (0 ≤ r < b)

a = 4q + r

Where r = 0, 1, 2, 3

When r = 1

a = 4q + 1 is a odd number

When r = 3

a = 4q + 3 which is also an odd number

So, we can say that any odd integer is in the form 4q + 1 or 4q + 3 where q is any integer.

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