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25 June, 06:06

5. Use Euclid's division lemma to show that the cube of any positive integer is of the form

9m, 9m + 1 or 9m+8.

2 The Fundamental Theorem of Arithmetic

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  1. 25 June, 07:09
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    according to euclids division lemma

    a=bq+r

    Step-by-step explanation:

    so, b=3. r=0,1,2

    a=3q+r

    cubing both sides

    a³ = (3q+0) ³

    a=3q³

    a=9q³ where m = q³

    a=9m

    r=1

    a³ = (3q+1) ³ (a+b) ³=a³+b³+3a²b+3ab²)

    a³ = (3q) ³+1³+3 (3q) ² (1) + 3 (3q) (1) ²

    =27q³+1+9q²+9q

    take common

    =9 (3q³+q²+q) + 1. where (3q³+q²+q) = m

    =9m+1

    r=2

    a³ = (3q+2) ³

    = (3q) ³+2³+3 (3q) ²+2) + 3 (3q) (2) ²

    =27q³+8+54q²+36q

    taking common

    =9 (3q³+6q²+4q) + 8 where (3q³+6q²+4q) = m

    9m + 8
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