NCERT Solutions for Ganita Prakash (Part 1) Chapter 1 A Square and A Cube
Updated on 2026-09-05
About this chapter
A locker is toggled once for each factor of its number, so it ends up open only when the factor count is odd. Factors come in partner pairs, and a pair repeats itself only in a square — so exactly the square-numbered lockers stay open. A perfect square is n × n = n 2 for a natural number n. Perfect squares end only in 0, 1, 4, 5, 6 or 9, and can carry only an even number of terminal zeros. The sum of the first n odd numbers is n 2 . Subtracting 1, 3, 5, … in turn is therefore a test: if you land exactly on 0 at the k-th step, the number is k 2 . Prime factorisation settles both questions. A number is a perfect square when its prime factors split into two identical groups, and a perfect cube when they split into three. Square root is the inverse of squaring; every perfect square has two int
- Queen Ratnamanjuri's puzzle
- Factors and partner factors
- Square Numbers
- Patterns and Properties of Perfect Squares
- Perfect Squares and Odd Numbers
- Perfect Squares and Triangular Numbers; Square Roots
- Square Roots
- Square Roots by prime factorisation
- –12Section 1.2 Cubic Numbers
- Cubic Numbers
Exercises
- Queen Ratnamanjuri's puzzle — In-text Questions Page 1
- Factors and partner factors — In-text Questions Page 2
- In-text Questions — Square Numbers Page 3
- Patterns and Properties of Perfect Squares — In-text Questions Page 4
- Patterns and Properties of Perfect Squares — In-text Questions Page 5
- Perfect Squares and Odd Numbers — In-text Questions Page 6
- Perfect Squares and Triangular Numbers; Square Roots — In-text Questions Page 7
- Square Roots — In-text Questions Page 8
- Square Roots by prime factorisation — In-text Questions Page 9
- Figure it Out — Square Numbers Page 10
- –12Section 1.2 Cubic Numbers — In-text Questions Page 11
- In-text Questions — Cubic Numbers Page 13
- Taxicab Numbers — Try This Page 13
- Perfect Cubes and Consecutive Odd Numbers; Cube Roots — In-text Questions Page 14
- Cube Roots; Successive Differences — In-text Questions Page 15
- In-text Questions — A Pinch of History Page 16
- Figure it Out — A Pinch of History Page 16
- Square Pairs! — Puzzle Time — Square Pairs! Page 18