A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer
Solution:
No.
Consider a as any positive integer of the form 3q + 1, where q is a natural number.
Here, a = 3q + 1
By squaring on both sides,
a2 = (3 q + 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 is an integer
Therefore, the square of a positive integer of the form 3q + 1 is always in the form 3m + 1 for some integer m
✦ Try This: Prove that square of any positive integer is of the form 4q or 4q + 1 for some integer q
Let 'a' be any positive integer.
b = 4
a = bq + r
a2 = (bq + r)2 --- 1.
r = 0, 1, 2, 3
From 1.
For r = 0,
a2 = (4q + 0)2
a2 = 16q2
a2 = 4(4q2)
= 4q, where q = 4q2.
for r = 1,
a2 = (4q + 1)2
a2 = 16q2 + 1 + 8q
a2 = 4(4q2 + 2q) + 1
a2 = 4q + 1, where q = 4q2 + 2q
Therefore, the square of any positive integer is of the form 4q or 4q + 1 for some integer q
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.2 Problem 5
A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer
Summary:
A positive integer is of the form 3q + 1, q being a natural number. The square of a positive integer of a form 3q + 1 is always in the form 3m + 1 for some integer m
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