# By taking three different, values of n verify the truth of the following statements:(i) If n is even, then n3 is also even.(ii) If n is odd, then n3 is also odd.(ii) If n leaves remainder 1 when divided by 3, then n3 also leaves 1 as remainder when divided by 3.(iv) If a natural number n is of the form 3p+2 then n3 also a number of the same type.

(i) If n is even, then n3 is also even.

Let the three even natural numbers be 2 , 4 , 6

Cubes of these numbers ,

= 23 = 8

= 43 = 64

= 63 = 216

Hence, we can see that all cubes are even in nature.

Statement verified.

(ii) If n is odd, then n3 is also odd.

Let three odd natural numbers are = 3 , 5 , 7

Cubes of these numbers =

= 33 = 27

= 53 = 125

= 73 = 343

Hence, we can see that all cubes are odd in nature.

Statement verified.

(iii) If n leaves remainder 1 when divided by 3, then n3 also leaves 1 as remainder when divided by 3.

Let three natural numbers of the form (3n+1) are = 4, 7 , 10

Cube of numbers = 43 = 64 , 73 = 343 , 103 = 1000

We can see that if we divide these numbers by 3 , we get 1 as remainder in each case.

Statement verified.

(iv) If a natural number n is of the form 3p+2 then n3 also a number of the same type.

Let three natural numbers of the form (3p+2) are = 5 , 8 , 11

Cube of these numbers = 53 = 125 , 83 = 512 , 113 = 1331

Now, we try to write these cubes in form of (3p + 2)

= 125 = 3 × 41 + 2

= 512 = 3 × 170 + 2

= 1331 = 3 × 443 + 2

Hence, statement verified.

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