Question

Write cubes of 4 natural numbers which are of the form 3n + 1 (e.g.

4, 7, 10, .........) and verify the following: 'The cube of a natural

number of the form 3n + 1 is a natural number of the same form'. [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks
Verification: 1 Mark

The 4 natural numbers which are of the form 3n + 1 (e.g. 4, 7, 10, .....) are as follows:

$3\times 1+1=3+1=4$

$3\times 2+1=6+1=7$

$3\times 3+1=9+1=10$

$3\times 4+1=12+1=13$

The cubes of 4 natural numbers which are of the form 3n + 1 (e.g

4, 7, 10, .........) are as follows:

$(4{)}^3=4\times 4\times 4=64$

$(7{)}^3=7\times 7\times 7=343$

$(10{)}^3=10\times 10\times 10=1000$

$(13{)}^3=13\times 13\times 13=2197$

Verfication:

$64=3\times 21+1$

$343=3\times 114+1$

$1000=3\times 333+1$

$2197=3\times 732+1$

$\therefore$ 'The cube of a natural number of the form 3n + 1 is a

natural number of the same form.'