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.