By factoring the expression, check that n³ - n is always divisible by 6 for all natural Class 9

By factoring the expression, check that n³ - n is always divisible by 6 for all natural Class 9

Question 1.

By factoring the expression, check that n³ - n is always divisible by 6 for all natural numbers n. Give reasons. Class 9

Solution:

n³ - n = n(n² -1)

= n(n + 1 )(n - 1)

...[∵ a² - b² = (a + b)(a -b)]

= (n- 1 )n(n + 1)

(n - 1), n, (n + 1) are three consecutive natural numbers.

Among any three consecutive integers:

i. At least one is divisible by 2

ii. One of them is divisible by 3

∴ (n - 1)n(n + 1) is divisible by both 2 and 3.

Hence, it is divisible by 2 × 3 = 6

Therefore, n³ - n is always divisible by 6 for all natural numbers n.