NCERT Solutions Ganita Prakash (Part 1) Chapter 1 In-text Questions — Cubic Numbers

Book page 13 Updated on2026-09-05

Q1.

We know that 0, 1, 4, 5, 6, 9 are the only last digits possible for squares. What are the possible last digits of cubes?

Answer

All ten digits, 0 to 9, are possible.

Last digit of the number	0	1	2	3	4	5	6	7	8	9
Last digit of its cube	0	1	8	7	4	5	6	3	2	9

Why cubes differ from squares: Each of the ten digits gives a different last digit when cubed, so the map is one-to-one and covers all ten. With squares, 2 and 8 both give 4, 3 and 7 both give 9, and so on — the map folds two digits onto one and only six outcomes survive. That is why the units digit rules out many non-squares but rules out no cube at all.

Tip: This makes guessing cube roots easy. 4913 ends in 3, so its cube root ends in 7; the root is between 10 and 20 because 10³ = 1000 and 20³ = 8000, so it must be 17.

Q2.

Similar to squares, can you find the number of cubes with 1 digit, 2 digits, and 3 digits? What do you observe?

Answer

Count them directly from 1³, 2³, 3³, … and compare with the squares.

Digits	Cubes	How many	Squares, for comparison	How many
1	1, 8	2	1, 4, 9	3
2	27, 64	2	16, 25, 36, 49, 64, 81	6
3	125, 216, 343, 512, 729	5	100, 121, …, 961	22

Observation: cubes are much rarer than squares. There are only 9 cubes below 1000, against 31 squares.

Why the thinning happens: The gap between consecutive cubes is (n+1)³ – n³ = 3n² + 3n + 1, which grows like n², while the gap between consecutive squares is only 2n + 1. Since the cube gaps widen far faster, fewer of them fit into any fixed stretch of numbers.

Q3.

Can a cube end with exactly two zeroes (00)? Explain.

Answer

No. A cube can end with 0 zeros, 3 zeros, 6 zeros … but never exactly 2.

10³ = 1000 → 3 zeros
20³ = 8000 → 3 zeros
100³ = 1000000 → 6 zeros

Why the count must be a multiple of 3: Write the number as m × 10^k where m does not end in 0. Cubing gives m³ × 10^3k, and m³ cannot end in 0 either — if it did, 10 would divide m³, forcing both 2 and 5 into m. So the cube ends in exactly 3k zeros, always a multiple of 3. Two zeros is impossible.

Tip: Compare with squares, where the terminal zeros always come in an even count. In each case the count of zeros is multiplied by the power you are taking.

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