FEUCAT Mathematics — Arithmetic — Multiples, Factors, PEMDAS, Fractions & DecimalsCheat Sheet
A printable cheat sheet for Arithmetic — Multiples, Factors, PEMDAS, Fractions & Decimals, built for FEUCAT reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the Far Eastern University-specific twists you will see on FEUCAT day.
Exam context
For the Far Eastern University College Admission Test, Far Eastern University tests Mathematics under a "Core section" label, with Arithmetic — Multiples, Factors, PEMDAS, Fractions & Decimals in the 1st slot across 9 chapters. FEUCAT candidates must clear the Competitive overall score cut on the 2026 paper, which draws about a meaningful share of Mathematics questions. Date to watch: Q3–Q4 2026.
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
Sections
Formulas
Formula
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
Formula
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
Section Title
Multiples and Factors
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
Term
Multiple
Example
25 is a multiple of 5 because 25 ÷ 5 = 5
Definition
A number that can be divided by another number without remainder
Term
Factor
Example
6 is a factor of 24 because 24 ÷ 6 = 4
Definition
A number that divides another number evenly (without remainder)
Term
Prime Number
Example
7 is prime (factors: 1, 7)
Definition
A number with exactly two factors: 1 and itself
Term
Composite Number
Example
12 is composite (factors: 1, 2, 3, 4, 6, 12)
Definition
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
Note
Example: 100 = 2² × 5²
Equation
Prime Factorization: Express any composite number as a product of prime factors
Conditions
Use factor trees or division method
Formulas
Formula
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
Formula
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
Formula
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
Section Title
LCM and GCF
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
Term
LCM (Least Common Multiple)
Example
LCM of 6 and 8 is 24
Definition
Smallest positive integer divisible by all given numbers
Term
GCF (Greatest Common Factor)
Example
GCF of 12 and 18 is 6
Definition
Largest positive integer that divides all given numbers
Term
Relatively Prime
Example
9 and 16 are relatively prime (GCF = 1)
Definition
Two numbers whose GCF is 1
Diagrams To Know
- Prime factorization trees
- Listing method for small numbers
Section Title
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
Term
Divisibility by 2
Example
246 is divisible by 2
Definition
Number ends in 0, 2, 4, 6, or 8
Term
Divisibility by 3
Example
123: 1+2+3=6, divisible by 3
Definition
Sum of digits is divisible by 3
Term
Divisibility by 4
Example
1,324: 24 ÷ 4 = 6
Definition
Last two digits form a number divisible by 4
Term
Divisibility by 5
Example
235 and 240 are divisible by 5
Definition
Number ends in 0 or 5
Term
Divisibility by 6
Example
48: even and 4+8=12 (divisible by 3)
Definition
Divisible by both 2 and 3
Term
Divisibility by 8
Example
5,216: 216 ÷ 8 = 27
Definition
Last three digits form a number divisible by 8
Term
Divisibility by 9
Example
729: 7+2+9=18, 1+8=9
Definition
Sum of digits is divisible by 9
Term
Divisibility by 10
Example
3,570 is divisible by 10
Definition
Number ends in 0
Formulas
Formula
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
Section Title
PEMDAS/Order of 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
Term
PEMDAS
Example
2 + 3 × 4 = 2 + 12 = 14 (not 20)
Definition
Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Term
Grouping Symbols
Example
[(2 + 3) × 4] - 1 = [5 × 4] - 1 = 19
Definition
Parentheses (), brackets [], braces {}
Diagrams To Know
- Step-by-step breakdown of complex expressions
Reactions Or Equations
Note
Common mistake: doing operations left to right without considering priority
Equation
Example: 3 + 2 × (4 + 1)² = 3 + 2 × 5² = 3 + 2 × 25 = 3 + 50 = 53
Conditions
Follow PEMDAS strictly
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Section Title
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
Term
Proper Fraction
Example
3/7, 5/8
Definition
Numerator smaller than denominator
Term
Improper Fraction
Example
7/3, 9/4
Definition
Numerator greater than or equal to denominator
Term
Mixed Number
Example
2¼ means 2 + 1/4
Definition
Whole number plus a proper fraction
Term
Equivalent Fractions
Example
1/2 = 2/4 = 3/6
Definition
Fractions that represent the same value
Diagrams To Know
- Fraction circles and bars for visualization
- Number line placement of fractions
Reactions Or Equations
Note
If ad > bc, then a/b > c/d
Equation
Cross multiplication for comparison: a/b vs c/d → compare ad and bc
Conditions
When comparing fractions without common denominator
Formulas
Formula
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
Formula
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
Section Title
Decimals
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
Term
Terminating Decimal
Example
0.25, 0.375
Definition
Decimal that ends (finite number of digits)
Term
Repeating Decimal
Example
0.333... = 0.3̄, 0.142857142857... = 0.142857̄
Definition
Decimal with repeating pattern of digits
Term
Place Value
Example
3.456: 3=ones, 4=tenths, 5=hundredths, 6=thousandths
Definition
Position of digit determines its value
Diagrams To Know
- Place value charts
- Decimal number lines
Reactions Or Equations
Note
For multiple repeating digits, use 99, 999, etc.
Equation
Repeating decimal to fraction: 0.7̄ = 7/9, 0.142857̄ = 1/7
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
Rows
Values
- Exactly 2
- More than 2
Property
Number of factors
Values
- 1 and itself
- 1, itself, and others
Property
Factors
Values
- 2, 3, 5, 7, 11
- 4, 6, 8, 9, 10
Property
Examples
Values
- 2 is only even prime
- 4 is smallest composite
Property
Special cases
Columns
- Property
- Prime
- Composite
Table Title
Prime vs Composite Numbers
Rows
Values
- List multiples, find smallest common
- List factors, find largest common
Property
Listing
Values
- Take highest power of each prime
- Take lowest power of common primes
Property
Prime factorization
Values
- Product of numbers
- Always 1
Property
For coprime numbers
Values
- LCM × GCF = product of numbers
- LCM × GCF = product of numbers
Property
Verification
Columns
- Method
- LCM
- GCF
Table Title
LCM vs GCF Methods
Rows
Values
- Numerator < denominator
- 3/7
- Already in simplest form
Property
Proper
Values
- Numerator ≥ denominator
- 7/3
- Convert to mixed number
Property
Improper
Values
- Whole + proper fraction
- 2¼
- Convert to improper for calculations
Property
Mixed
Values
- Numerator = 1
- 1/5
- Building block for other fractions
Property
Unit
Columns
- Type
- Definition
- Example
- How to use
Table Title
Fraction Types
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