use Euclid division algorithm to show that the cube of any positive integer is of the form 4m or 4m + 1 or 4m+3
Answers
Step-by-step explanation:
Let a be an arbitrary positive integer. Then, by Euclid’s division algorithm, corresponding to the positive integers a and 4, there exist non-negative integers q and r such that a = 4q + r, where 0< r< 4
a = 4q + r, where 0 ≤ r < 4
⇒ a³ = (4q + r)³ = 64q³ + r³ + 12qr² + 48q²r
[∵(a+b)³ = a³ + b³ + 3ab2 + 3a2b]
⇒ a³ = (64q² + 48q²r + 12qr²) + r³
where, 0 ≤ r < 4
Case I
When r = 0,
Putting r = 0 in Eq.(i), we get
a³ = 64q³ = 4(16q³)
⇒ a³= 4m where m = 16q³ is an integer.
Case II
When r = 1, then putting r = 1 in Eq.(i), we get
a³ = 64q³ + 48q² + 12q + 1
= 4(16q³ + 12q² + 3q) + 1
= 4m + 1
where, m = (16q³ + 12q² + 3q) is an integer.
Case II
When r = 2, then putting r = 2 in Eq.(i), we get
a³ = 64q³ + 144q² + 108q + 27
= 64q³ + 144q² + 108q + 24 + 3
= 4(16q³ + 36q² + 27q + 6) + 3 = 4m + 3
where, m = (16q³ + 36q² + 27q + 6) is an integer.
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