Integers, PEMDAS & Divisibility - Cheat Sheet
Your last-minute revision companion for mastering integers, order of operations, and divisibility rules for all major Philippine entrance exams
Integers and Number Properties
Formulas
|n| = n if n ≥ 0; |n| = -n if n < 0
Meaning
Absolute value of any number n
Watch out
Remember |−5| = 5, not −5
When to use
When finding distance from zero or removing negative signs
Common values
Value
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Symbol
P₁₀
Quantity
First 10 primes
Important facts
- Zero (0) is neither positive nor negative
- Zero is even but neither prime nor composite
- 1 is neither prime nor composite
- 2 is the only even prime number
- Any number × 0 = 0
- Division by zero is undefined
- 0 ÷ (any non-zero number) = 0
Key definitions
Integer
Example
−7, 0, 15 are integers; 2.5, 1/3 are not
Whole numbers including positive, negative, and zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}
Absolute Value
Example
|−8| = 8, |5| = 5
The positive representation of a number, denoted by |n|
Prime Number
Example
2, 3, 5, 7, 11, 13, 17, 19, 23
Number greater than 1 divisible only by 1 and itself
Composite Number
Example
4, 6, 8, 9, 10, 12, 14, 15, 16
Number greater than 1 with more than two factors
Diagrams to know
- Number line showing positive and negative integers
- Factor tree for prime factorization
Operations with Integers (Law of Signs)
Formulas
Addition: Same signs → add, keep sign; Different signs → subtract, take sign of larger
Meaning
Rule for adding positive and negative integers
Watch out
−3 + 5 = 2 (not 8), because you subtract and take the sign of 5
When to use
When adding any two integers
Multiplication/Division: Same signs → positive; Different signs → negative
Meaning
Sign rules for multiplication and division
Watch out
−4 × −3 = +12 (positive because same signs)
When to use
When multiplying or dividing integers
Important facts
- Positive + Positive = Positive
- Negative + Negative = Negative (add magnitudes)
- Positive + Negative = Sign of larger magnitude
- Same signs multiply/divide → Positive result
- Different signs multiply/divide → Negative result
- Subtraction = Addition of opposite
Key definitions
Subtraction Rule
Example
7 − (−3) = 7 + 3 = 10
Change subtraction to addition of the opposite: a − b = a + (−b)
Diagrams to know
- Sign rules chart for all four operations
Reactions or equations
(+a) + (+b) = +(a + b)
Note
Result is positive
Conditions
Both numbers positive
(−a) + (−b) = −(a + b)
Note
Add magnitudes, result is negative
Conditions
Both numbers negative
(+a) × (−b) = −(a × b)
Note
Result is always negative
Conditions
Different signs
(−a) × (−b) = +(a × b)
Note
Result is always positive
Conditions
Both negative
PEMDAS - Order of Operations
Formulas
PEMDAS: Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right)
Meaning
Correct order for evaluating mathematical expressions
Watch out
MD and AS have equal priority - work left to right within each pair
When to use
Any expression with multiple operations
Important facts
- Always work from innermost parentheses outward
- Exponents are evaluated right to left: 2^3^2 = 2^9 = 512
- Multiplication and division have equal priority
- Addition and subtraction have equal priority
- Implied multiplication: 3(4) = 3 × 4
- Any number^0 = 1 (except 0^0 which is indeterminate)
Key definitions
PEMDAS
Example
2 + 3 × 4 = 2 + 12 = 14 (not 20)
Parentheses, Exponents, Multiplication & Division, Addition & Subtraction
Grouping Symbols
Example
2 + [3 × (4 + 1)] = 2 + [3 × 5] = 2 + 15 = 17
Parentheses ( ) innermost, then brackets [ ], then braces { }
Diagrams to know
- PEMDAS hierarchy pyramid
- Step-by-step solution flowchart
Reactions or equations
2 − [4 − 3 × (−3 + 5)²] + 4 ÷ 2 × 6 − 3 = 19
Note
Work parentheses first, then exponents, then left to right
Conditions
Step-by-step PEMDAS application
Factors and Multiples
Formulas
GCF by Prime Factorization: Multiply common prime factors with lowest powers
Meaning
Method to find Greatest Common Factor
Watch out
Use lowest powers of common primes only
When to use
When finding GCF of two or more numbers
LCM by Prime Factorization: Multiply all prime factors with highest powers
Meaning
Method to find Least Common Multiple
Watch out
Use highest powers of all primes that appear
When to use
When finding LCM of two or more numbers
Important facts
- 1 is a factor of every number
- Every number is a factor of itself
- Prime numbers have exactly two factors: 1 and themselves
- Composite numbers have more than two factors
- Every number is a multiple of itself
- Multiples are infinite, factors are finite
- GCF is always ≤ smallest number
- LCM is always ≥ largest number
Key definitions
Factor
Example
Factors of 12: 1, 2, 3, 4, 6, 12
A number that divides another number evenly (no remainder)
Multiple
Example
Multiples of 5: 5, 10, 15, 20, 25, ...
Result of multiplying a number by any integer
GCF
Example
GCF(18, 24) = 6
Greatest Common Factor - largest number that divides all given numbers
LCM
Example
LCM(4, 6) = 12
Least Common Multiple - smallest positive multiple common to all given numbers
Diagrams to know
- Factor tree diagram
- Venn diagram for finding GCF/LCM
Reactions or equations
18 = 2 × 3²; 24 = 2³ × 3
Note
GCF = 2¹ × 3¹ = 6; LCM = 2³ × 3² = 72
Conditions
Prime factorization
Divisibility Rules
Important facts
- Divisible by 2: Last digit is 0, 2, 4, 6, or 8
- Divisible by 3: Sum of digits is divisible by 3
- Divisible by 4: Last two digits form a number divisible by 4
- Divisible by 5: Last digit is 0 or 5
- Divisible by 6: Divisible by both 2 and 3
- Divisible by 8: Last three digits form a number divisible by 8
- Divisible by 9: Sum of digits is divisible by 9
- Divisible by 10: Last digit is 0
- Divisible by 11: Alternating sum of digits is divisible by 11
- Divisible by 12: Divisible by both 3 and 4
Key definitions
Divisibility
Example
20 is divisible by 4 because 20 ÷ 4 = 5 with no remainder
A number divides another evenly (remainder = 0)
Diagrams to know
- Divisibility rules flowchart
- Quick reference table for all rules
Scientific Notation
Formulas
a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer
Meaning
Standard form for scientific notation
Watch out
Coefficient must be between 1 and 10 (not including 10)
When to use
For very large or very small numbers
Important facts
- Positive exponent: large number (move decimal right)
- Negative exponent: small number (move decimal left)
- Count decimal places moved to determine exponent
- Coefficient must be 1 ≤ |a| < 10
- For calculations: convert to decimal, compute, convert back
Key definitions
Scientific Notation
Example
0.0045 = 4.5 × 10⁻³; 12,300 = 1.23 × 10⁴
Way to express numbers as a × 10ⁿ where 1 ≤ |a| < 10
Diagrams to know
- Decimal point movement diagram
- Powers of 10 reference chart
Reactions or equations
34,000 = 3.4 × 10⁴
Note
Move decimal 4 places left, exponent is +4
Conditions
Converting large number
0.000023 = 2.3 × 10⁻⁵
Note
Move decimal 5 places right, exponent is -5
Conditions
Converting small number
Must remember
- PEMDAS order: Parentheses, Exponents, MD (left to right), AS (left to right)
- Same signs in multiplication/division = Positive result
- Different signs in multiplication/division = Negative result
- Divisible by 3: sum of digits divisible by 3
- Divisible by 9: sum of digits divisible by 9
- Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
- Zero is even but neither positive nor negative
- Any number × 0 = 0, but division by 0 is undefined
- Scientific notation: 1 ≤ |coefficient| < 10
- Absolute value is always positive: |−5| = 5
Last minute tips
- For PEMDAS problems, write each step clearly - don't skip steps mentally
- When checking divisibility by 3 or 9, add digits until you get a single digit
- Remember: multiplication and division are equal priority (work left to right)
- For negative numbers, use parentheses to avoid sign errors: (−3) × (−4) = +12
- Quick factor check: if sum of digits is divisible by 3, the whole number is too
Comparison tables
Prime vs Composite Numbers
| Property | Prime Numbers | Composite Numbers |
|---|---|---|
| Definition | Exactly 2 factors | More than 2 factors |
| Factors | 1 and itself only | 1, itself, and others |
| Examples | 2, 3, 5, 7, 11 | 4, 6, 8, 9, 10 |
| Smallest | 2 | 4 |
| Even examples | 2 only | 4, 6, 8, 10, 12... |
Operations Sign Rules
| Operation | Same Signs | Different Signs |
|---|---|---|
| Addition | Add, keep sign | Subtract, sign of larger |
| Subtraction | Convert to addition | Convert to addition |
| Multiplication | Positive result | Negative result |
| Division | Positive result | Negative result |
Quick Divisibility Check
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit even | 124 (ends in 4) |
| 3 | Sum of digits ÷ 3 | 123 → 1+2+3=6 ÷ 3 |
| 5 | Ends in 0 or 5 | 125, 130 |
| 9 | Sum of digits ÷ 9 | 729 → 7+2+9=18 ÷ 9 |
| 10 | Ends in 0 | 120, 340 |
Diagrams
Integers and Operations Overview
Comprehensive overview of integer types, operations, and special properties
PEMDAS Solution Process
Step-by-step flowchart for applying PEMDAS rules to solve mathematical expressions
Divisibility Rules Quick Check
Quick decision tree for checking common divisibility rules