Show that any positive integer is of the form 3q or, 3q+1 or,3q+2 for some integer q.

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Solution

Let a be any positive integer and b=3. Applying division lemma with a and b=3, we have
a=3q+r ,where,0 is less than or equal to r and r is less than 3 and q is some integer.
a=3q+0 or,3q+1 or, 3q+2
a=3q or,a=3q+1 or, a=3q+2 for some integer q.

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Q. Question 4
Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
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Say true or false:

A positive integer is of the form 3q + 1, q being a natural number, then you write its square in any form other than 3m + 1, i.e.,3m or 3m + 2 for some integer m.

Q. A positive integer is of the form

3 q + 1, q

being a anatural number. Can you write its square in any form other than

3 m + 1, 3 m

3 m + 2

for some integer

m

? Justify your answer.

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Show that any positive integer is of the form 3q or, 3q+1 or,3q+2 for some integer q.

Let a be any positive integer and b=3. Applying division lemma with a and b=3, we havea=3q+r ,where,0 is less than or equal to r and r is less than 3 and q is some integer.a=3q+0 or,3q+1 or, 3q+2a=3q or,a=3q+1 or, a=3q+2 for some integer q.

Let a be any positive integer and b=3. Applying division lemma with a and b=3, we have
a=3q+r ,where,0 is less than or equal to r and r is less than 3 and q is some integer.
a=3q+0 or,3q+1 or, 3q+2
a=3q or,a=3q+1 or, a=3q+2 for some integer q.