Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
Solution:
Let a be the positive integer and b = 4.
Then, by Euclid’s algorithm, a = 4q + r for some integer q ≥ 0 and r = 0, 1, 2, 3 because 0 ≤ r < 4.
So, a = 4q or 4q + 1 or 4q + 2 or 4q + 3.
(4q)3 = 64q3 = 4(16q3)
= 4m, where m is some integer.
(4q + 1)3 = 64q3 + 48q2 + 12q + 1
= 4(16q3 + 12q2 + 3) + 1
= 4m + 1, where m is some integer.
(4q + 2)3 = 64q3 + 96q2 + 48q + 8
= 4(16q3 + 24q2 + 12q + 2)
= 4m, where m is some integer.
(4q + 3)3 = 64q3 + 144q2 + 108q + 27
= 4(16q3 + 36q2 + 27q + 6) + 3
= 4m + 3, where m is some integer.
Hence, The cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for some integer m.
Let a be any positive integer and b = 4.
According to Euclid Division Lemma,
a = bq + r [0 ≤ r < b]
a = 3q + r [0 ≤ r < 4]
According to the question, the possible values of r are,
r = 0, r = 1, r = 2, r = 3
When r = 0,
a = 4q + 0
a = 4q
Taking cubes on LHS and RHS,
We have,
a³ = (4q)³
a³ = 4 (16q³)
a³ = 9m [where m is an integer = 16q³]
When r = 1,
a = 4q + 1
Taking cubes on LHS and RHS,
We have,
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]
When r = 2,
a = 4q + 2
Taking cubes on LHS and RHS,
We have,
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]
When r = 3,
a = 4q + 3
Taking cubes on LHS and RHS,
We have,
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]
Hence, the cube of any positive integer is in the form of 4m, 4m+1 or 4m+3.