Types of Numbers

Natural Numbers

1, 2, 3, 4, 5, ...

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Whole Numbers

0, 1, 2, 3, 4, ...

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Integers

..., -3, -2, -1, 0, 1, 2, 3, ...

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Even Numbers

2, 4, 6, 8, ...

General Form:

$$2n$$

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Odd Numbers

1, 3, 5, 7, ...

General Form:

$$2n+1$$

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Prime Numbers

Memorize prime numbers up to 50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47

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Composite Numbers

Non-prime numbers greater than 1.

Examples:

4, 6, 8, 9, 10, 12

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Important Results

Sum of First n Natural Numbers

$$1+2+\cdots+n ============ \frac{n(n+1)}{2}$$

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Sum of Squares

$$1^2+2^2+\cdots+n^2 ================== \frac{n(n+1)(2n+1)}{6}$$

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Sum of Cubes

$$1^3+2^3+\cdots+n^3 ================== \left(\frac{n(n+1)}{2}\right)^2$$

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2. DIVISIBILITY RULES

Must memorize.

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Divisible by 2

Last digit is:

0, 2, 4, 6, 8

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Divisible by 3

Sum of digits is divisible by 3.

Example:

123

1 + 2 + 3 = 6

Since 6 is divisible by 3, 123 is divisible by 3.

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Divisible by 4

Last two digits are divisible by 4.

Example:

1316

16 ÷ 4 = 4

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Divisible by 5

Last digit is:

0 or 5

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Divisible by 6

The number is divisible by both:

2 and 3

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Divisible by 8

Last three digits are divisible by 8.

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Divisible by 9

Sum of digits is divisible by 9.

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Divisible by 10

Last digit is:

0

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Divisible by 11

Difference between alternating digit sums is:

0 or a multiple of 11

Example:

121

(1 + 1) − 2 = 0

Therefore, 121 is divisible by 11.

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3. HCF & LCM

HCF (Highest Common Factor)

Example:

12, 18

HCF:

6

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LCM (Least Common Multiple)

Example:

12, 18

LCM:

36

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Most Important Formula

For two numbers only:

$$\text{HCF}\times\text{LCM} ========================== \text{Product of the Numbers}$$

Example:

$$12\times18=216$$

Also,

$$6\times36=216$$

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Fractions Shortcut

HCF of Fractions

$$\frac{\text{HCF of Numerators}} {\text{LCM of Denominators}}$$

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LCM of Fractions

$$\frac{\text{LCM of Numerators}} {\text{HCF of Denominators}}$$

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4. REMAINDERS

Frequently asked.

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Basic Rule

$$\text{Dividend} =============== (\text{Divisor}\times\text{Quotient}) + \text{Remainder}$$

or

$$N=DQ+R$$

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Remainder Range

$$0\le R<\text{Divisor}$$

Always.

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Common Shortcut

Find the remainder when:

17 ÷ 5

Answer:

2

Because:

$$17=(5\times3)+2$$

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Large Powers

Last digits occur in cycles.

Example:

$$2^1=2$$ $$2^2=4$$ $$2^3=8$$ $$2^4=16$$

Last digit pattern:

2, 4, 8, 6

Cycle length:

4

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Important Last Digit Cycles

2

2, 4, 8, 6

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3

3, 9, 7, 1

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4

4, 6

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7

7, 9, 3, 1

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8

8, 4, 2, 6

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9

9, 1

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5. DECIMALS & FRACTIONS

Decimal to Fraction

0.5   = 1/2
0.25  = 1/4
0.75  = 3/4

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Important Values

0.125 = 1/8
0.2   = 1/5
0.4   = 2/5
0.625 = 5/8

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Fraction Comparison Shortcut

Cross multiply.

Example:

Compare:

$$\frac{3}{7} \quad\text{and}\quad \frac{4}{9}$$

Cross products:

$$3\times9=27$$ $$4\times7=28$$

Since:

$$28>27$$

Therefore,

$$\frac{4}{9}>\frac{3}{7}$$

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6. SURDS & INDICES

Indices Laws

Multiplication

$$a^m\cdot a^n ============ a^{m+n}$$

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Division

$$\frac{a^m}{a^n} =============== a^{m-n}$$

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Power of a Power

$$(a^m)^n ======= a^{mn}$$

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Zero Power

$$a^0=1 \qquad (a\neq0)$$

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Negative Power

$$a^{-n} ====== \frac{1}{a^n}$$

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Surds

Numbers involving roots.

Examples:

√2

√3

√5

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Rationalization

Most common form:

$$\frac{1}{\sqrt{a}} ================== \frac{\sqrt{a}}{a} \qquad (a>0)$$

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7. LOGARITHMS

Definition

$$\log_a(a^n)=n$$

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Important Rules

Product Rule

$$\log(ab) ======== \log a+\log b$$

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Quotient Rule

$$\log\left(\frac{a}{b}\right) ============================ \log a-\log b$$

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Power Rule

$$\log(a^n) ========= n\log a$$

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Log of 1

$$\log_a1=0$$

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Log of Base

$$\log_aa=1$$

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8. AP (ARITHMETIC PROGRESSION)

Example:

2, 5, 8, 11, 14, ...

Common Difference:

3

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nth Term

$$a_n === a+(n-1)d$$

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Sum of n Terms

$$S_n === \frac{n}{2}\left[2a+(n-1)d\right]$$

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9. GP (GEOMETRIC PROGRESSION)

Example:

2, 6, 18, 54, ...

Common Ratio:

3

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nth Term

$$a_n === ar^{n-1}$$

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Sum of n Terms

$$S_n === a\left(\frac{r^n-1}{r-1}\right) \qquad (r\neq1)$$

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Infinite GP

$$S_\infty ======== \frac{a}{1-r}$$

where

$$|r|<1$$

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10. HP (HARMONIC PROGRESSION)

Definition

A sequence is in HP if its reciprocals form an AP.

Example:

1, 1/2, 1/3, 1/4, ...

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Relationship

HP → Reciprocals form an AP

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MOST IMPORTANT FORMULAS

Sum of Natural Numbers

$$\frac{n(n+1)}{2}$$

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HCF × LCM

$$\text{HCF}\times\text{LCM} ========================== \text{Product}$$

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AP nth Term

$$a+(n-1)d$$

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GP nth Term

$$ar^{n-1}$$

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Indices

$$a^m\cdot a^n ============ a^{m+n}$$

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Logarithm

$$\log(ab) ======== \log a+\log b$$