Show that :

(i) 13³ - 5³ is divisible by 8.

(ii) 35³ + 27³ is divisible by 62.

(i) We know that

a³ - b³ = (a - b)(a² + ab + b²)

Factorising 13³ - 5³, we get :

⇒ 13³ - 5³ = (13 - 5)[13² + 13 × 5 + 5²]

= 8(169 + 65 + 25)

= 8 × 259, which is divisible by 8.

Hence, proved that 13³ - 5³ is divisible by 8.

(ii) We know that

a³ + b³ = (a + b)(a² - ab + b²)

Factorising 35³ + 27³, we get :

⇒ 35³ + 27³ = (35 + 27)[(35)² - 35 × 27 + (27)²]

= 62[1225 - 945 + 729]

= 62 × 1009, which is divisible by 62.

Hence, proved that 35³ + 27³ is divisible by 62.