Question

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

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

Answer 1

Three natural numbers of the form (3n + 1) can be written by choosing \[n = 1, 2, 3 . . . etc.\]

Let three such numbers be  \[4, 7 \text{ and }  10 .\]

Cubes of the three chosen numbers are:

\[4^3 = 64, 7^3 = 343 \text{ and  } {10}^3 = 1000\] Cubes of \[4, 7 \text{ and } 10\] can expressed as: \[64 = 3 \times 21 + 1\], which is of the form (3n + 1) for n = 21 

\[343 = 3 \times 114 + 1\] , which is of the form (3n + 1) for n = 114

\[1000 = 3 \times 333 + 1,\]  which is of the form (3n + 1) for n = 333 Cubes of  \[4, 7, \text{ and } 10\] can be expressed as the natural numbers of the form (3n + 1) for some natural number n. Hence, the statement is verified.