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Question

Show that any positive integer is in the form of 3q,3q+1, 3q+1for some integer q


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Solution

Euclid's division lemma:

Let us consider a positive integer a. We apply here Euclid's division algorithm a=bq+r (where a is dividend bis divisor ,q is the quotient and r is remainder). We have a and b=3.

Since 0r<3(because remainder cannot be greater than the divisor). Possible remainders in this case are 0, 1and 2.

So, three cases arises according to Eucild's division Lemma a=bq+r

First case: When r=0,

a=3q+0

a=3q

Second case:When r=1,

a=3q+1

Third case:When r=2,

a=3q+2

Thus,any positive integer is in the form of 3q,3q+1, 3q+1for some integer.


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