Civil Service Exam (Subprofessional) Numerical Ability — Integers, PEMDAS & DivisibilityCheat Sheet
One-page cheat sheet for Civil Service Exam (Subprofessional) Numerical Ability — Integers, PEMDAS & Divisibility. Every formula, definition, and key fact you need for this chapter, condensed to a single printable page. Designed for the final review session before the Civil Service Exam (Subprofessional) 2026.
Exam context
The Career Service Examination — Subprofessional Level is conducted by Civil Service Commission (CSC) and is scheduled for Bi-annual — March and August 2026. The Numerical Ability subtest is marked as "~25% weightage" in the official pattern, and Integers, PEMDAS & Divisibility appears in position 1st of 9 in the Civil Service Exam (Subprofessional) Numerical Ability review rotation. Passing mark: 80%. Recent Civil Service Exam (Subprofessional) 2026 papers have drawn roughly 17 questions from this subject.
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
Sections
Formulas
Formula
|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
Section Title
Integers and Number Properties
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
Term
Integer
Example
−7, 0, 15 are integers; 2.5, 1/3 are not
Definition
Whole numbers including positive, negative, and zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}
Term
Absolute Value
Example
|−8| = 8, |5| = 5
Definition
The positive representation of a number, denoted by |n|
Term
Prime Number
Example
2, 3, 5, 7, 11, 13, 17, 19, 23
Definition
Number greater than 1 divisible only by 1 and itself
Term
Composite Number
Example
4, 6, 8, 9, 10, 12, 14, 15, 16
Definition
Number greater than 1 with more than two factors
Diagrams To Know
- Number line showing positive and negative integers
- Factor tree for prime factorization
Formulas
Formula
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
Formula
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
Section Title
Operations with Integers (Law of Signs)
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
Term
Subtraction Rule
Example
7 − (−3) = 7 + 3 = 10
Definition
Change subtraction to addition of the opposite: a − b = a + (−b)
Diagrams To Know
- Sign rules chart for all four operations
Reactions Or Equations
Note
Result is positive
Equation
(+a) + (+b) = +(a + b)
Conditions
Both numbers positive
Note
Add magnitudes, result is negative
Equation
(−a) + (−b) = −(a + b)
Conditions
Both numbers negative
Note
Result is always negative
Equation
(+a) × (−b) = −(a × b)
Conditions
Different signs
Note
Result is always positive
Equation
(−a) × (−b) = +(a × b)
Conditions
Both negative
Formulas
Formula
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
Section Title
PEMDAS - Order of 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
Term
PEMDAS
Example
2 + 3 × 4 = 2 + 12 = 14 (not 20)
Definition
Parentheses, Exponents, Multiplication & Division, Addition & Subtraction
Term
Grouping Symbols
Example
2 + [3 × (4 + 1)] = 2 + [3 × 5] = 2 + 15 = 17
Definition
Parentheses ( ) innermost, then brackets [ ], then braces { }
Diagrams To Know
- PEMDAS hierarchy pyramid
- Step-by-step solution flowchart
Reactions Or Equations
Note
Work parentheses first, then exponents, then left to right
Equation
2 − [4 − 3 × (−3 + 5)²] + 4 ÷ 2 × 6 − 3 = 19
Conditions
Step-by-step PEMDAS application
Formulas
Formula
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
Formula
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
Section Title
Factors and Multiples
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
Term
Factor
Example
Factors of 12: 1, 2, 3, 4, 6, 12
Definition
A number that divides another number evenly (no remainder)
Term
Multiple
Example
Multiples of 5: 5, 10, 15, 20, 25, ...
Definition
Result of multiplying a number by any integer
Term
GCF
Example
GCF(18, 24) = 6
Definition
Greatest Common Factor - largest number that divides all given numbers
Term
LCM
Example
LCM(4, 6) = 12
Definition
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
Note
GCF = 2¹ × 3¹ = 6; LCM = 2³ × 3² = 72
Equation
18 = 2 × 3²; 24 = 2³ × 3
Conditions
Prime factorization
Section Title
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
Term
Divisibility
Example
20 is divisible by 4 because 20 ÷ 4 = 5 with no remainder
Definition
A number divides another evenly (remainder = 0)
Diagrams To Know
- Divisibility rules flowchart
- Quick reference table for all rules
Formulas
Formula
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
Section Title
Scientific Notation
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
Term
Scientific Notation
Example
0.0045 = 4.5 × 10⁻³; 12,300 = 1.23 × 10⁴
Definition
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
Note
Move decimal 4 places left, exponent is +4
Equation
34,000 = 3.4 × 10⁴
Conditions
Converting large number
Note
Move decimal 5 places right, exponent is -5
Equation
0.000023 = 2.3 × 10⁻⁵
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
Rows
Values
- Exactly 2 factors
- More than 2 factors
Property
Definition
Values
- 1 and itself only
- 1, itself, and others
Property
Factors
Values
- 2, 3, 5, 7, 11
- 4, 6, 8, 9, 10
Property
Examples
Values
- 2
- 4
Property
Smallest
Values
- 2 only
- 4, 6, 8, 10, 12...
Property
Even examples
Columns
- Property
- Prime Numbers
- Composite Numbers
Table Title
Prime vs Composite Numbers
Rows
Values
- Add, keep sign
- Subtract, sign of larger
Property
Addition
Values
- Convert to addition
- Convert to addition
Property
Subtraction
Values
- Positive result
- Negative result
Property
Multiplication
Values
- Positive result
- Negative result
Property
Division
Columns
- Operation
- Same Signs
- Different Signs
Table Title
Operations Sign Rules
Rows
Values
- Last digit even
- 124 (ends in 4)
Property
2
Values
- Sum of digits ÷ 3
- 123 → 1+2+3=6 ÷ 3
Property
3
Values
- Ends in 0 or 5
- 125, 130
Property
5
Values
- Sum of digits ÷ 9
- 729 → 7+2+9=18 ÷ 9
Property
9
Values
- Ends in 0
- 120, 340
Property
10
Columns
- Divisor
- Rule
- Example
Table Title
Quick Divisibility Check
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