Stste that any odd positive integers …
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Posted by Kavitha Kombiah 5 years, 8 months ago
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Gaurav Seth 5 years, 8 months ago
Let n be a positive odd integer. We need to show that n can be written in any one of the form of 8q+1, 8q+3, 8q+5 or 8q+7
According to division algorithm,
we can write any number ‘a’ in the form
a = 8q + r
where q is any integer and 0 <= r <= 7. So r can be 0, 1, 2, 3, 4, 5, 6 or 7.
Thus, a can be written as
a = 8q
a = 8q+2
a = 8q+3
a = 8q+4
a = 8q+5
a = 8q+6
a = 8q+7
We need only odd numbers. Since 8q, 8q+2, 8q+4, and 8q+6 are divisible by 2, they are even numbers.
So any odd integer can be written as any one of the remaining forms which are (8q+1, 8q+3, 8q+5 or 8q+7.).
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