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.
[Hint: Let x be any positive integer then it is of the form
. Now square each of these and show that they can be rewritten in the form 3m or 3m+1]
Solution:
Let x be any positive integer and b=3.
According to Euclid’s division lemma, we can say that
Therefore, all possible values of x are:
or
Now lets square each one of them one by one.
(i)
Let
be some integer, we get
(ii)
Let
be some integer, we get
(iii)
Let
be some integer, we get
Hence, square of any positive integer is either of the form 3m or 3m+1 for some integer m.
Singh says
Oh Bhai 6 khaha se aa gaya 2nd part me