Question

Show that cube of any positive integer is of the form 3p,3p+1or3p+8by using Euclid's division lemma

Answer 1

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It must be like this ,

$\underline { \red { Question }}$

Show that the cube of any positive integer is of the form 9p, 9p+1 or 9p+8.

$\underline { \pink { Solution:}}$

Let 'a' be any positive integer.

/* we apply the division lemma ,with a and b = 3 .*/

Since , 0 ≤ r < 3 , the possible remainders are 0, 1 and 2 .

a can be 3m, or 3m+1 or 3m+2 , where m is the quotient.

Now, ( 3m )³ = 27m³

= 9( 3m³ )

= 9p , [where p = 3m³ ] ---(1)

Again , ( 3m + 1 )³

= 27m³ + 27m² + 9m + 1

= 9( 3m³ + 9m² + 1 ) + 1

= 9p + 1 , [where p = 3m³ + 9m² + 1] ---(2)

Lastly, ( 3m + 8 )³

= 27m³ + 54m² + 36m + 8

= 9( 3m³ + 6m² + 4m) + 8

= 9p + 8 , [where p = 3m³ + 6m² + 4m ] ---(3)

Therefore.,

The cube of any positive integer is of the form 9p, 9p+ 1 or 9p + 8 .

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