Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer
Solution:
No,the square of any positive integer cannot be written in the form 3m + 2 where m is a natural number.
From the Euclid’s division lemma, we know that
A positive integer ‘a’ can be written in the form bq + r
a = bq + r,
where b, q and r are any integers,
For b = 3
a = 3(q) + r,
where, r can be an integers,
For r = 0, 1, 2, 3……….
3q + 0, 3q + 1, 3q + 2, 3q + 3……. are positive integers,
(3q)2 = 9q2
= 3(3q2)
= 3m (where 3q2 = m)
(3q + 1)2 = (3q + 1)2
Using the algebraic identity (a + b)2 = a2 + b2 + 2ab
= 9q2 + 1 + 6q
Taking out 3 as common
= 3(3q2 + 2q) + 1
= 3m + 1 (Where, m = 3q2 + 2q)
(3q + 2)2 = (3q + 2)2
Using the algebraic identity (a + b)2 = a2 + b2 + 2ab
= 9q2 + 4 + 12q
Taking out 3 as common
= 3(3q2 + 4q) + 4
= 3m + 4 (Where, m = 3q2 + 2q)
(3q + 3)2 = (3q + 3)2
Using the algebraic identity (a + b)2 = a2 + b2 + 2ab
= 9q2 + 9 + 18q
Taking out 3 as common
= 3(3q2 + 6q) + 9
= 3m + 9 (Where, m = 3q2 + 2q)
Therefore, the square of any positive integer cannot be of the form 3m + 2
✦ Try This: Write whether the square of any positive integer can be of the form 2m + 1, where m is a natural number. Justify your answer
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.2 Problem 4
Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer
Summary:
The square of any positive integer cannot be of the form 3m + 2, where m is a natural number.
☛ Related Questions:
visual curriculum