Question

By taking three different values of n verify the truth of the following statement:

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

Answer 1

Three natural numbers of the form (3p + 2) can be written by choosing $$p=1,2,3...etc.$$

Let three such numbers be  $$5,8\text{ and }11.$$ 

Cubes of the three chosen numbers are:  $$5^3=125,8^3=512\text{ and }{11}^3=1331$$ Cubes of  $$5,8,\text{ and }11$$  can be expressed as: $$125=3\times 41+2$$, which is of the form (3p + 2) for p = 41 $$512=3\times 170+2$$, which is of the form (3p + 2) for p = 170 $$1331=3\times 443+2,$$ which is of the form (3p + 2) for p = 443
Cubes of $$5,8,\text{ and }11$$  could be expressed as the natural numbers of the form (3p + 2) for some natural number p. Hence, the statement is verified.