Factoring n cube – n gives the expression (n – 1)(n)(n + 1), which represents three consecutive integers. In any three consecutive numbers, at least one is a multiple of 2 and one is a multiple of 3.
By factoring the expression, check that n cube – n is always divisible by 6 for all natural numbers n.
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We begin by completely factoring the given cubic expression. Taking out the common term n gives n(n square – 1), which further opens up into the three-part continuous variable expression (n – 1)(n)(n + 1). This product represents three consecutive natural numbers. In any sequence of three consecutive numbers, at least one number must be even (divisible by 2) and exactly one must be a multiple of 3. Their product is therefore always divisible by 2 times 3, which equals 6.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-4/