NCERT Solutions for Class 8th Maths Chapter 5 Number Play
Updated on 2026-09-19
About this chapter
Placing '+' and '–' signs between four numbers gives 8 expressions, and all 8 have the same parity. Switching one sign changes the value by 2 × (that number), an even change — so the parity can never change. The same idea, one level up: even numbers split into 4p and 4p + 2. Two of the same type add to a multiple of 4; one of each type never does. 'Multiple of 4' behaves exactly like 'even' does inside the whole numbers. Four divisibility facts are established and used throughout: if a | M and a | N then a | M + N and a | M – N; if a | A then a divides no more than the multiples of A that k allows — precisely, all multiples of A are divisible by a; if A is divisible by k then A is divisible by every factor of k; and if A is divisible by both k and m then A is divisible by LCM(k, m). Every
- .1 Is This a Multiple Of?
- Breaking Even
- Breaking Even / Pairs to Make Fours
- Pairs to Make Fours
- Always, Sometimes, or Never
- Always, Sometimes, or Never / What Remains?
- What Remains?
- .2 Checking Divisibility Quickly
- A Shortcut for Divisibility by 9
- A Shortcut for Divisibility by 3
Quick revision
| Divisor | Shortcut | Why it works | Example |
|---|---|---|---|
| 10 | Units digit is 0 | Every place value above the units is a multiple of 10 | 4070 ✓, 4075 ✗ |
| 5 | Units digit is 0 or 5 | 10, 100, 1000 … are all multiples of 5 | 4075 ✓, 4073 ✗ |
| 2 | Units digit is even | 10, 100, 1000 … are all multiples of 2 | 708 ✓, 477 ✗ |
| 4 | Last two digits form a multiple of 4 | 100, 1000, 10000 … are multiples of 4 | 2856 → 56 ✓ |
| 8 | Last three digits form a multiple of 8 | 1000, 10000 … are multiples of 8 | 2856 → 856 = 8 × 107 ✓ |
| 3 | Digit sum is a multiple of 3 | 9, 99, 999 … are multiples of 3, so N and its digit sum leave the same remainder | 429714 → 27 ✓ |
| 9 | Digit sum is a multiple of 9 | 1 = 0 + 1, 10 = 9 + 1, 100 = 99 + 1, … | 405 → 9 ✓, 8888 → 32 ✗ |
| 11 | (sum of digits in the 1, 100, 10000 places) – (sum of digits in the 10, 1000 places) is 0 or a multiple of 11 | Place values are alternately 1 more and 1 less than a multiple of 11 | 462 → 4 – 6 + 2 = 0 ✓ |
| 6 | Divisible by 2 and by 3 | 2 and 3 are coprime, LCM (2, 3) = 6 | 186 ✓, 225 ✗ |
| 24 | Divisible by 3 and by 8 (not by 4 and 6) | 24 = 2³ × 3; LCM (4, 6) is only 12 | 12 passes 4 and 6 but fails 24 |
| Digital root | Add digits repeatedly to one digit | Adding digits never changes the remainder mod 9 | 489710 → 29 → 11 → 2 |
Exercises
- .1 Is This a Multiple Of? — In-text Questions Page 1125
- .1 Is This a Multiple Of? — In-text Questions Page 1135
- .1 Is This a Multiple Of? — In-text Questions Page 1145
- Breaking Even — In-text Questions Page 115
- Breaking Even / Pairs to Make Fours — In-text Questions Page 116
- Pairs to Make Fours — In-text Questions Page 117
- Always, Sometimes, or Never — In-text Questions Page 118
- Always, Sometimes, or Never — In-text Questions Page 119
- Always, Sometimes, or Never — In-text Questions Page 120
- Always, Sometimes, or Never / What Remains? — In-text Questions Page 121
- What Remains? — In-text Questions Page 122
- .1 Is This a Multiple Of? — Figure it Out Page 1225
- .2 Checking Divisibility Quickly — In-text Questions Page 1235
- A Shortcut for Divisibility by 9 — In-text Questions Page 124
- A Shortcut for Divisibility by 9 — In-text Questions Page 125
- A Shortcut for Divisibility by 9 — Figure it Out Page 126
- A Shortcut for Divisibility by 3 — In-text Questions Page 126
- A Shortcut for Divisibility by 11 — In-text Questions Page 127
- A Shortcut for Divisibility by 11 — In-text Questions Page 128
- A Shortcut for Divisibility by 11 / More on Divisibility Shortcuts — In-text Questions Page 129
- Divisibility Shortcuts for Other Numbers / Digital Roots — In-text Questions Page 130
- Digital Roots — Figure it Out Page 131
- .3 Digits in Disguise — In-text Questions Page 1315
- .3 Digits in Disguise — In-text Questions Page 1325
- .3 Digits in Disguise — Figure it Out Page 1325