Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.
Answers & Comments
brainly11422
Let a be any positive integer. Then by Euclid’s division lemma, we have a = bq + r, where 0 ≤ r < b For b = 3, we have a = 3q + r, where 0 ≤ r < 3 ...(i)
So, The numbers are of the form 3q, 3q + 1 and 3q + 2. So, (3q)2 = 9q2 = 3(3q2) = 3m, where m is a integer. (3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1, where m is a integer. (3q + 2)2 = 9q2 + 12q + 4, which cannot be expressed in the form 3m + 2. Therefore, Square of any positive integer cannot be expressed in the form 3m + 2.
9 votes Thanks 11
SINGHisKING11
Agle din phir jb lekhak aaya aur dekha ki aaj unka chamah change ho Gaya h. usne paan wale se pucha ki ye kaise pratima h jo Roz roz chamah badalata h. To paan wala muh se paan thookte hue kehte h ki wo hi desh bhakt lagaya h
Answers & Comments
a = bq + r, where 0 ≤ r < b
For b = 3, we have
a = 3q + r, where 0 ≤ r < 3 ...(i)
So, The numbers are of the form 3q, 3q + 1 and 3q + 2.
So, (3q)2 = 9q2 = 3(3q2)
= 3m, where m is a integer.
(3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1
= 3m + 1,
where m is a integer.
(3q + 2)2 = 9q2 + 12q + 4,
which cannot be expressed in the form 3m + 2.
Therefore, Square of any positive integer cannot be expressed in the form 3m + 2.