Types of numbers, divisibility rules, LCM and HCF, BODMAS, surds and indices, with worked examples and practice
This is the arithmetic base for the whole quantitative section. Master divisibility, LCM and HCF, BODMAS order, and the laws of indices first, because percentage ratio and average, profit loss and interest, and time speed distance and time and work all sit on top of them.
| Type | Definition | Examples |
|---|---|---|
| Natural | Counting numbers from 1 | 1, 2, 3, ... |
| Whole | Natural numbers plus 0 | 0, 1, 2, ... |
| Integers | Whole numbers plus their negatives | ..., -2, -1, 0, 1, 2, ... |
| Rational | Can be written as p/q, q not 0 | 3, -5, 2/7, 0.25, 0.333... |
| Irrational | Non-terminating, non-repeating decimal | √(2), π, e |
| Prime | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11, 13 |
| Composite | More than two factors | 4, 6, 8, 9, 10 |
| Co-prime | Two numbers with HCF 1 | (8, 15), (9, 16) |
Note: 1 is neither prime nor composite. 2 is the only even prime. There are 25 primes below 100.
| Divisor | Rule |
|---|---|
| 2 | Last digit even (0, 2, 4, 6, 8) |
| 3 | Digit sum divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Digit sum divisible by 9 |
| 10 | Last digit 0 |
| 11 | Difference of (sum of odd-place digits) and (sum of even-place digits) is 0 or a multiple of 11 |
| Concept | Rule |
|---|---|
| LCM times HCF | LCM(a, b) times HCF(a, b) = a times b (for two numbers) |
| Sum of first n naturals | n(n+1)/2 |
| Sum of first n squares | n(n+1)(2n+1)/6 |
| Sum of first n cubes | [n(n+1)/2] squared |
| (a+b) squared | a2 + 2ab + b2 |
| (a-b) squared | a2 - 2ab + b2 |
| a2 - b2 | (a+b)(a-b) |
| a3 + b3 | (a+b)(a2 - ab + b2) |
| a3 - b3 | (a-b)(a2 + ab + b2) |
| Law | Form |
|---|---|
| Product | am times an = am+n |
| Quotient | am / an = am-n |
| Power of power | (am)n = amn |
| Zero power | a0 = 1 (a not 0) |
| Negative power | a-n = 1 / an |
| Fractional power | am/n = nth root of (am) |
A surd is an irrational root such as √(5). To rationalise a denominator, multiply numerator and denominator by the conjugate: for 1/(a + √(b)) multiply by (a - √(b)).
Brackets, Orders (powers and roots), Division, Multiplication, Addition, Subtraction. Division and multiplication rank equally and are done left to right; the same holds for addition and subtraction. Inside brackets, the innermost is resolved first.
Simplify 12 + 6 of 3 - 8 / 4 times 2.
"of" behaves like multiplication and is done with the bracket-level priority, so 6 of 3 = 18. Now 12 + 18 - 8 / 4 times 2. Division and multiplication left to right: 8 / 4 = 2, then 2 times 2 = 4. So 12 + 18 - 4 = 26.
The HCF of two numbers is 12 and their LCM is 360. If one number is 60, find the other.
Use LCM times HCF = product of the numbers. 360 times 12 = 60 times other. Other = 4320 / 60 = 72.
Is 91827 divisible by 11?
Odd-place digits (from right: 7, 8, 9) sum to 24. Even-place digits (2, 1) sum to 3. Difference = 24 - 3 = 21, which is not 0 or a multiple of 11. So 91827 is not divisible by 11.
Find 1 + 2 + 3 + ... + 50.
Use n(n+1)/2 with n = 50: 50 times 51 / 2 = 2550 / 2 = 1275.
Simplify (34 times 32) / 33.
Numerator: 34+2 = 36. Then 36 / 33 = 36-3 = 33 = 27.
Rationalise 1 / (3 + √(2)).
Multiply top and bottom by the conjugate (3 - √(2)). Denominator becomes 32 - (√(2))2 = 9 - 2 = 7. Result = (3 - √(2)) / 7.