Question

show that any odd integer is of the form 8k+1​

Answer 1

We have

Any positive integer is of the form 8k+1.

As per Euclid’s Division lemma.

If a and b are two positive integers, then,

a=bq+r

Where 0≤r<b.

Let positive integers be a and b=8

Hence,a=bq+r

Where, (0≤r<8)

R is an integer greater than or equal to 0 and less than 8

Hence, r can be either 0,1,2,3,4,5,6,7 and 8.

Now, If r=1

Then, the equation becomes

a=bq+r

a=8q+1

This will always be odd integer.

Hence proved.

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