use Euclid's division lemma to show that the square of any positive integers is of the form 3p,3p+1
Answers
your answer is in the attached picture
at place of n put any required value and according to your ques. you put p at place of n
ok !
and
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Answer:
If a and b are two positive integers, then,
a = bq + r, 0 ≤ r ≤ b
Let b = 3
Therefore, r = 0, 1, 2
Therefore, a = 3q or a = 3q + 1 or a = 3q + 2
(i). If a = 3q then,
⇒ a² = 9q² = 3(3q²)
= 3m
where m = 3q²
(ii). If a = 3q + 1 then,
⇒ a² = 9q² + 6q + 1 = 3(3q + 2q) + 1
= 3p + 1
where p = 3q² + 2q
(iii). If a = 3q + 2
⇒ a² = 9q² + 12q + 4 = 3(3q² + 4q + 1) + 1
= 3p + 1,
where m = 3q² + 4q + 1
Therefore, the square of any positive integer is either of the form 3p or 3p + 1.
hope u will be helped by this.