Question

Show that any positive odd integer can be written in the form 6 m + 1, 6 m + 3 or 6

m + 5 where m is a positive integer.

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Answer 1

SoluTioπ :-

Let take a as any positive integer and b = 6. a > b

Then using Euclid’s algorithm,

a = 6q + r

r is remainder

value of q is ≥ 0

r = 0, 1, 2, 3, 4, 5 because 0 ≤ r < b

value of b is 6

So total possible forms will be 6q + 0, 6q + 1 , 6q + 2,6q + 3,6q + 4,6q + 5.

6q + 0 → 6 is divisible by 2 so it is an even number.

6q + 1 → 6 is divisible by 2 but 1 is not divisible by 2 so it is an odd number.

6q + 2 → 6 is divisible by 2 and 2 is also divisible by 2 so it is an even number.

6q + 3 → 6 is divisible by 2 but 3 is not divisible by 2 so it is an odd number.

6q + 4 → 6 is divisible by 2 and 4 is also divisible by 2 it is an even number.

6q + 5 → 6 is divisible by 2 but 5 is not divisible by 2 so it is an odd number.

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5.

$\sf {\rule {120}{2} \ Be\ Brainly\ \star }$