Arithmetic — Multiples, Factors, PEMDAS, Fractions & Decimals - Cheat sheet
Your last-minute revision companion for mastering arithmetic fundamentals — all key formulas, rules, and problem-solving strategies in one place
Multiples and Factors
Formulas
If m = n × k (where k is a positive integer), then m is a multiple of n
Meaning
m = multiple, n = original number, k = multiplier
Watch out
Don't confuse multiples with factors — multiples are larger than or equal to the original number
When to use
When determining if one number is a multiple of another
If n ÷ m = whole number with remainder 0, then m is a factor of n
Meaning
n = larger number, m = potential factor
Watch out
Remember that 1 and the number itself are always factors
When to use
When finding all factors of a number
Common values
Value
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Symbol
P₁₀
Quantity
First 10 primes
Important facts
- 2 is the only even prime number
- 0 and 1 are neither prime nor composite
- If m is a factor of n, then n is a multiple of m
- Prime numbers from 2 to 20: 2, 3, 5, 7, 11, 13, 17, 19
Key definitions
Multiple
Example
25 is a multiple of 5 because 25 ÷ 5 = 5
A number that can be divided by another number without remainder
Factor
Example
6 is a factor of 24 because 24 ÷ 6 = 4
A number that divides another number evenly (without remainder)
Prime Number
Example
7 is prime (factors: 1, 7)
A number with exactly two factors: 1 and itself
Composite Number
Example
12 is composite (factors: 1, 2, 3, 4, 6, 12)
A number with more than two factors
Diagrams to know
- Factor trees for prime factorization
- Venn diagrams for common factors and multiples
Reactions or equations
Prime Factorization: Express any composite number as a product of prime factors
Note
Example: 100 = 2² × 5²
Conditions
Use factor trees or division method
LCM and GCF
Formulas
LCM × GCF = Product of the two numbers
Meaning
LCM = Least Common Multiple, GCF = Greatest Common Factor
Watch out
This only works for exactly two numbers, not three or more
When to use
To verify your LCM and GCF calculations
LCM by prime factorization: Take highest power of each prime factor
Meaning
Combine all prime factors using their maximum exponents
Watch out
Don't forget to include ALL prime factors from both numbers
When to use
When finding LCM of large numbers
GCF by prime factorization: Take lowest power of common prime factors only
Meaning
Use only primes that appear in both factorizations with minimum exponent
Watch out
Only use prime factors that appear in BOTH numbers
When to use
When finding GCF of large numbers
Important facts
- If one number divides another, the smaller is the GCF and larger is the LCM
- GCF is always ≤ smaller number; LCM is always ≥ larger number
- For relatively prime numbers: GCF = 1, LCM = product of numbers
Key definitions
LCM (Least Common Multiple)
Example
LCM of 6 and 8 is 24
Smallest positive integer divisible by all given numbers
GCF (Greatest Common Factor)
Example
GCF of 12 and 18 is 6
Largest positive integer that divides all given numbers
Relatively Prime
Example
9 and 16 are relatively prime (GCF = 1)
Two numbers whose GCF is 1
Diagrams to know
- Prime factorization trees
- Listing method for small numbers
Divisibility Rules
Important facts
- For divisibility by 7: Double last digit, subtract from remaining number
- You can repeat digit sum process until single digit for rules 3 and 9
- Divisibility by 6 requires BOTH rules for 2 and 3 to be satisfied
Key definitions
Divisibility by 2
Example
246 is divisible by 2
Number ends in 0, 2, 4, 6, or 8
Divisibility by 3
Example
123: 1+2+3=6, divisible by 3
Sum of digits is divisible by 3
Divisibility by 4
Example
1,324: 24 ÷ 4 = 6
Last two digits form a number divisible by 4
Divisibility by 5
Example
235 and 240 are divisible by 5
Number ends in 0 or 5
Divisibility by 6
Example
48: even and 4+8=12 (divisible by 3)
Divisible by both 2 and 3
Divisibility by 8
Example
5,216: 216 ÷ 8 = 27
Last three digits form a number divisible by 8
Divisibility by 9
Example
729: 7+2+9=18, 1+8=9
Sum of digits is divisible by 9
Divisibility by 10
Example
3,570 is divisible by 10
Number ends in 0
PEMDAS/Order of Operations
Formulas
PEMDAS: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
Meaning
P=(), E=powers, MD=×÷ (left to right), AS=+- (left to right)
Watch out
Multiplication and Division have EQUAL priority (work left to right)
When to use
For any expression with multiple operations
Important facts
- Always work from innermost grouping symbols outward
- Multiplication and Division have equal priority (work left to right)
- Addition and Subtraction have equal priority (work left to right)
- Exponents are evaluated right to left: 2³² = 2⁹ = 512
Key definitions
PEMDAS
Example
2 + 3 × 4 = 2 + 12 = 14 (not 20)
Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Grouping Symbols
Example
[(2 + 3) × 4] - 1 = [5 × 4] - 1 = 19
Parentheses (), brackets [], braces {}
Diagrams to know
- Step-by-step breakdown of complex expressions
Reactions or equations
Example: 3 + 2 × (4 + 1)² = 3 + 2 × 5² = 3 + 2 × 25 = 3 + 50 = 53
Note
Common mistake: doing operations left to right without considering priority
Conditions
Follow PEMDAS strictly
Fractions
Formulas
Addition/Subtraction: a/b ± c/d = (ad ± bc)/(bd)
Meaning
Find common denominator, then add/subtract numerators
Watch out
Don't add denominators directly — find LCM first
When to use
When adding or subtracting fractions with different denominators
Multiplication: a/b × c/d = (a×c)/(b×d)
Meaning
Multiply numerators together, multiply denominators together
Watch out
Simplify by canceling common factors before multiplying
When to use
When multiplying any fractions
Division: a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)
Meaning
Multiply by the reciprocal of the divisor
Watch out
Don't flip the wrong fraction — flip the second one (divisor)
When to use
When dividing fractions
Mixed number to improper: a(b/c) = (a×c + b)/c
Meaning
Multiply whole number by denominator, add numerator
Watch out
Don't forget to add the original numerator
When to use
Converting mixed numbers for calculations
Common values
Value
1/2 = 0.5, 1/3 ≈ 0.333, 1/4 = 0.25, 3/4 = 0.75
Symbol
Decimal equivalents
Quantity
Common fractions
Important facts
- Always simplify fractions to lowest terms
- To compare fractions, find common denominator or cross multiply
- When adding/subtracting, find LCM of denominators
- Reciprocal of a/b is b/a
Key definitions
Proper Fraction
Example
3/7, 5/8
Numerator smaller than denominator
Improper Fraction
Example
7/3, 9/4
Numerator greater than or equal to denominator
Mixed Number
Example
2¼ means 2 + 1/4
Whole number plus a proper fraction
Equivalent Fractions
Example
1/2 = 2/4 = 3/6
Fractions that represent the same value
Diagrams to know
- Fraction circles and bars for visualization
- Number line placement of fractions
Reactions or equations
Cross multiplication for comparison: a/b vs c/d → compare ad and bc
Note
If ad > bc, then a/b > c/d
Conditions
When comparing fractions without common denominator
Decimals
Formulas
Fraction to decimal: Divide numerator by denominator
Meaning
Perform long division until remainder is 0 or pattern repeats
Watch out
Some fractions give repeating decimals — use bar notation
When to use
Converting fractions to decimal form
Decimal to fraction: Write over power of 10, then simplify
Meaning
0.75 = 75/100 = 3/4
Watch out
Count decimal places to determine denominator
When to use
Converting terminating decimals to fractions
Common values
Value
1/8 = 0.125, 1/6 ≈ 0.167, 2/3 ≈ 0.667, 5/8 = 0.625
Symbol
Memorize these
Quantity
Common decimal equivalents
Important facts
- When multiplying decimals, count total decimal places in factors
- When dividing by decimal, move decimal point in both numbers
- Comparing decimals: align decimal points and compare digit by digit
- Rounding: look at digit to the right of rounding place
Key definitions
Terminating Decimal
Example
0.25, 0.375
Decimal that ends (finite number of digits)
Repeating Decimal
Example
0.333... = 0.3̄, 0.142857142857... = 0.142857̄
Decimal with repeating pattern of digits
Place Value
Example
3.456: 3=ones, 4=tenths, 5=hundredths, 6=thousandths
Position of digit determines its value
Diagrams to know
- Place value charts
- Decimal number lines
Reactions or equations
Repeating decimal to fraction: 0.7̄ = 7/9, 0.142857̄ = 1/7
Note
For multiple repeating digits, use 99, 999, etc.
Conditions
For single repeating digit, use 9 as denominator
Must remember
- PEMDAS order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)
- LCM × GCF = product of the two numbers (verification formula)
- 2 is the only even prime number; 0 and 1 are neither prime nor composite
- When adding fractions: find common denominator first, don't add denominators
- Division of fractions: multiply by reciprocal of divisor (flip second fraction)
- Divisibility by 3 and 9: sum of digits must be divisible by 3 or 9 respectively
- Multiples are ≥ original number; factors are ≤ original number
- Prime factorization: every composite number has unique prime factorization
- For decimal to fraction: count decimal places to determine power of 10 in denominator
- Relatively prime means GCF = 1 (numbers share no common factors except 1)
Last minute tips
- For LCM/GCF problems: if one number divides the other, smaller = GCF, larger = LCM
- Quick divisibility check for 6: number must be even AND sum of digits divisible by 3
- When simplifying fractions, find GCF of numerator and denominator, then divide both
- For PEMDAS: if you see only multiplication and division, work strictly left to right
- Convert mixed numbers to improper fractions before doing any calculations
Comparison tables
Prime vs Composite Numbers
| Property | Prime | Composite |
|---|---|---|
| Number of factors | Exactly 2 | More than 2 |
| Factors | 1 and itself | 1, itself, and others |
| Examples | 2, 3, 5, 7, 11 | 4, 6, 8, 9, 10 |
| Special cases | 2 is only even prime | 4 is smallest composite |
LCM vs GCF Methods
| Method | LCM | GCF |
|---|---|---|
| Listing | List multiples, find smallest common | List factors, find largest common |
| Prime factorization | Take highest power of each prime | Take lowest power of common primes |
| For coprime numbers | Product of numbers | Always 1 |
| Verification | LCM × GCF = product of numbers | LCM × GCF = product of numbers |
Fraction Types
| Type | Definition | Example | How to use |
|---|---|---|---|
| Proper | Numerator < denominator | 3/7 | Already in simplest form |
| Improper | Numerator ≥ denominator | 7/3 | Convert to mixed number |
| Mixed | Whole + proper fraction | 2¼ | Convert to improper for calculations |
| Unit | Numerator = 1 | 1/5 | Building block for other fractions |
Diagrams
Arithmetic Concepts Overview
Complete overview of all arithmetic topics covered in this chapter, showing relationships between concepts
Prime or Composite Decision Tree
Step-by-step decision process to determine if a number is prime or composite
PEMDAS Problem Solving Steps
Systematic approach to solving expressions using proper order of operations
Finding LCM and GCF Strategy
Decision tree for choosing the best method to find LCM and GCF based on the size of numbers