Q. 225.0( 2 Votes )

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.

Answer :

(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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