Question

Show that the cube of any positive integers will be in the form of 8m+1 8m+3 8m+5 8m+7

Answer 1

⇒ Write first a few perfect cubes.

• 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,...

⇒ Divide each term by 8 and write down each remainder obtained.

• 1, 0, 3, 0, 5, 0, 7, 0, 1, 0, 3, 0, 5, 0, 7, 0,...

⇒ Here the remainders obtained are 0, 1, 3, 5 and 7.

⇒ The perfect cubes which leave remainder 0 can be written as 8m.

⇒ The perfect cubes which leave remainder 1 can be written as 8m + 1.

⇒ The perfect cubes which leave remainder 3 can be written as 8m + 3.

⇒ The perfect cubes which leave remainder 5 can be written as 8m + 5.

⇒ The perfect cubes which leave remainder 7 can be written as 8m + 7.

⇒ Hence proved!