show the cube of any positive integer will be in the form of 8m or 8m+1 or 8m+3 o r8m+5
Answers
AnswEr:
let us Consider that a & b are two positive integers.
By Using Euclid's Division Lemma
Here, a = bq + r where, 0 < r < b and r can be 0, 1, 3, 5.
a = 8q
Cubing Both Sides
↬ (a)³ = (8q)³
↬ a³ = 512q³
↬ a³ = 8 × (64q)³ [m = 64³]
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a = 8q + 1
Cubing Both Sides
↬ (a)³ = (8q + 1)³
↬ a³ = 8q³ + 1³ + 3(8q)²(1) + 3(8q) (1)²
↬ a³ = 512q³ + 192q² + 24q
- Taking Common 8
↬a³ = 8(64q³ + 24q² + 3q) + 1 [m = 64q³ + 24q2² + 3q]
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a = 8q + 3
Cubing Both Sides
↬(a)³ = (8q + 3)³
↬ a³ = 8q³ + 3² + 3(8q)² (3) + 3(8a) (3)²
↬ a = 512q³ + 27 + 576q² + 216q
- Taking Common 8
↬a³ = 8(64q + 72q² + 27q + 3) + 3
[m = (64q + 72q² + 27q + 3) ]
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a = 8q + 5
Cubing Both Sides
↬(a)³ = (8q + 5)³
↬a³ = (8q)³ + (5)³ + 3(8q)²(5) + 3(8q) (5)²
↬a³ = 513q³ + 125 + 960q² + 600q
↬a = 8(64³ + 120q² + 175q + 15) + 5 + 5
Here, [ m = (64³ + 120q² + 175q + 15) ]
Hence, the cube of any positive integer can be in the form of 8m or 8m+1 or 8m+3 o r8m+5.
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