Question

use division algorithm to show that the cube of any positive integer is of the form 9 m,9m+1or 9m + 8​

Answer 1

Step-by-step explanation:

According to the Euclid's division algorithm, a = bq + r (where 0 ≤ r < b)

r here can be 0,1,2,3,4,5,6,7 and 8.

Here, let a be any positive integer, and b = 9

• When r = 0

$\longrightarrow$ a = 9m

$\longrightarrow$ a³ = (9m)³ ---- (Cubing both sides)

$\longrightarrow$ a³ = 729m³

$\longrightarrow$ a³ = 9(81m³)

$\longrightarrow$ a³ = 9m [where m is 81m³]

___________________________

• When r = 1

$\longrightarrow$ a = 9m + 1

$\longrightarrow$ a³ = (9m + 1)³ ---- (Cubing both sides)

$\longrightarrow$ a³ = (9m)³ + (1)³ + 3(9m)(1)(9m + 1)

$\longrightarrow$ a³ = 729m³ + 1 + 27m(9m + 1)

$\longrightarrow$ a³ = 729m³ + 1 + 243m² + 27m

$\longrightarrow$ a³ = 9(81m³ + 27m² + 3) + 1

$\longrightarrow$ a³ = 9m + 1 [where m = (81m³ + 27m² + 3)]

___________________________

• When r = 8

$\longrightarrow$ a = 9m + 8

$\longrightarrow$ a³ = (9m + 8)³ ---- (Cubing both sides)

$\longrightarrow$ a³ = (9m)³ + (8)³ + 3(9m)(8)(9m + 8)

$\longrightarrow$ a³ = 729m³ + 512 + 216m(9m + 8)

$\longrightarrow$ a³ = 729m³ + 512 + 1944m² + 1728m

$\longrightarrow$ a³ = 9(81m³ + 216m² + 192m + 56) + 8

$\longrightarrow$ a³ = 9m + 8 [where m = (81m³ + 216m² + 192m + 56)]

Hence, it is proved that cube of any positive integer is of the form 9 m,9m+1or 9m + 8.

Answer 2

Given :-

9m

9m + 1

9m + 8

To Find :-

use division algorithm to show that the cube of any positive integer

Solution :-

Let x be the number

x = 9m

x³ = 9m³

x³ = 729m³

Now

729 can be written as 9(81)

x³ = 9m

2]

9m + 1

Now

r = 1

x³ = (9m + 1)³

Apply identity

(a + b)³ = a³ + b³ + 3ab(a + b)

x³ = (9m)³ + (1)³ + 3 × 9m × 1(9m + 1)

x³ = 729m³ + 1 + 3 × 9m × 1(9m + 1)

x³ = 729m³ + 1 + 27m(9m + 1)

x³ = 729m³ + 1 + 27m + 243m²

x³ = 9(81m³ + 27m² + 3) + 1

x³ = 9(m) + 1

x³ = 9m + 1

3]

x³ = (9m + 8)³

Using same identity

(a + b)³ = a³ + b³ + 3ab(a + b)

x³ = (9m)³ + (8)³ + 3(9m)(8)(9m + 8)

x³ = 729m³ + 512 + 27m(8)(9m + 8)

x³ = 729m³ + 512 + 1944m² + 1728m

x³ = 9(81m³ + 216m² + 192m + 56) + 8

x³ = 9(m) + 8

x³ = 9m + 8

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