Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m
Solution:
Consider a as any positive integer and b = 4.
From the Euclid’s division lemma,
a = bq + r [0 ≤ r < b]
a = 3q + r [0 ≤ r < 4]
Possible values of r are,
r = 0, r = 1, r = 2, r = 3
Case 1: When r = 0,
a = 4q + 0
a = 4q
On taking cubes on both sides LHS and RHS, we get,
a3 = (4q)3
a3= 4(16q3)
a3 = 4m [where m is an integer = 16q3]
Case 2: When r = 1,
a = 4q + 1
On taking cubes on both sides LHS and RHS, we get,
a3 = (4q + 1)3
a3 = 64q3 + 13 + 3 × 4q × 1 (4q + 1)
a3 = 64q3 + 1 + 48q2 + 12q
a3 = 4(16q3 + 12q2 + 3q) + 1
a3 = 4m + 1 [where m is an integer = 16q3 + 12q2 + 3q]
Case 3: When r = 2,
a = 4q + 2
On taking cubes on both sides LHS and RHS, we get,
a3 = (4q + 2)3
a3 = 64q3 + 23 + 3 × 4q × 2 (4q + 2)
a3 = 64q3 + 8 + 96q2 + 48q
a3 = 4 (16q3 + 2 + 24q2 + 12q)
a3 = 4m [where m is an integer =16q3 + 2 + 24q2 + 12q]
Case 4: When r = 3,
a = 4q + 3
On taking cubes on both sides LHS and RHS, we get,
a3 = (4q + 3)3
a3 = 64q3 + 27 + 3 × 4q × 3 (4q + 3)
a3 = 64q3 + 24 + 3 + 144q2 + 108q
a3 = 4(16q3 + 36q2 + 27q + 6) + 3
a3 = 4m + 3 [where m is an integer = 16q3 + 36q2 +27q + 6]
Therefore, the cube of any positive integer is in the form of 4m, 4m + 1 or 4m + 3
✦ Try This: Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.3 Problem 2
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m
Summary:
The cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m. Hence Proved
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