show that any positive integer is of the form 3q or 3q+1 or 3q+ 2 for some integer q
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Since you have to show that every positive integer is of form 3q, 3q + 1 or 3q + 2 where q is any integer, consider b = 3 and let a be any integer.
Then By Euclid's Algorithm, we can write
a = 3q + r
where q is some integer and r is the remainder.
As we are dividing a by 3, so the value of r cannot be more than 2.
So r = 0 or 1 or 2
∴ a = 3q + 0 or 3q + 1 or 3q + 2
⇒ a = 3q or 3q + 1 or 3q + 2
As a is a positive integer, so we can say that every positive integer is of form 3q, 3q + 1 or 3q + 2
Then By Euclid's Algorithm, we can write
a = 3q + r
where q is some integer and r is the remainder.
As we are dividing a by 3, so the value of r cannot be more than 2.
So r = 0 or 1 or 2
∴ a = 3q + 0 or 3q + 1 or 3q + 2
⇒ a = 3q or 3q + 1 or 3q + 2
As a is a positive integer, so we can say that every positive integer is of form 3q, 3q + 1 or 3q + 2
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