Arithmetic — Multiples, Factors, PEMDAS, Fractions & Decimals - Revision notes
This comprehensive review covers fundamental arithmetic concepts essential for UPCAT and other college entrance exams. Master these concepts through step-by-step problem solving, understand when and how to apply different techniques, and build confidence in tackling arithmetic problems efficiently.
Multiples and Factors
Formulas
If m is a factor of n, then n is a multiple of m
Example
Since 6 is a factor of 24, then 24 is a multiple of 6
Variables
m = factor, n = number being factored
Application
Used to identify factor-multiple relationships
Prime Factorization: n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ
Example
100 = 2² × 5² = 4 × 25
Variables
n = number, pᵢ = prime factors, aᵢ = exponents
Application
Express any composite number as product of prime factors
Exam tips
- Memorize prime numbers up to 50 for quick factorization
- Use factor trees systematically to avoid errors in prime factorization
- When finding factors, list them in pairs to ensure you don't miss any
Key points
- A multiple of n is any number that can be divided by n without remainder (e.g., 15, 20, 25 are multiples of 5)
- A factor of n is any number that divides n without remainder (e.g., factors of 12 are 1, 2, 3, 4, 6, 12)
- Prime numbers have exactly two factors: 1 and themselves (2, 3, 5, 7, 11, 13, 17, 19, 23...)
- Composite numbers have more than two factors (4, 6, 8, 9, 10, 12...)
- To test if a large number n is prime: check divisibility by all primes up to √n
Definitions
Prime Number
A natural number greater than 1 with exactly two factors: 1 and itself
Importance
Foundation for prime factorization and number theory problems
Composite Number
A natural number greater than 1 with more than two factors
Importance
Can be broken down into prime factors for easier computation
Common mistakes
- Confusing multiples with factors (remember: 24 is a multiple of 6, but 6 is a factor of 24)
- Forgetting that 1 is neither prime nor composite
- Not checking all prime numbers up to √n when testing for primality
LCM and GCF
Formulas
LCM(a,b) × GCF(a,b) = a × b
Example
For 12 and 18: LCM(36) × GCF(6) = 216 = 12 × 18 ✓
Variables
a, b = two numbers
Application
Check your LCM and GCF calculations
LCM = product of highest powers of all prime factors
Example
24 = 2³×3, 18 = 2×3²; LCM = 2³×3² = 72
Variables
prime factors from both numbers
Application
Find LCM using prime factorization
GCF = product of lowest powers of common prime factors
Example
24 = 2³×3, 18 = 2×3²; GCF = 2¹×3¹ = 6
Variables
common prime factors only
Application
Find GCF using prime factorization
Exam tips
- Always verify using the formula: LCM × GCF = product of numbers
- List method works well for small numbers, prime factorization for larger ones
- Practice identifying when numbers are relatively prime
Key points
- LCM (Least Common Multiple): smallest number divisible by both given numbers
- GCF (Greatest Common Factor): largest number that divides both given numbers
- LCM × GCF = Product of the two numbers
- For prime factorization method: LCM uses highest powers, GCF uses lowest powers
- If one number divides another, the smaller is GCF and larger is LCM
Definitions
Relatively Prime
Two numbers whose GCF is 1
Importance
Important for fraction simplification and probability problems
Common mistakes
- Using addition instead of multiplication when combining prime factors
- Forgetting to include all prime factors in LCM calculation
- Using highest powers for GCF instead of lowest powers
Divisibility Rules
Formulas
For divisibility by 3 or 9: repeatedly add digits until single digit
Example
2847: 2+8+4+7=21, 2+1=3, divisible by 3 but not 9
Variables
original number's digits
Application
Quick divisibility check without division
Exam tips
- Memorize rules for 2, 3, 4, 5, 6, 8, 9, 10 - they appear frequently
- Use divisibility rules to quickly eliminate answer choices
- Practice the digit sum method for large numbers
Key points
- Divisibility by 2: number ends in 0, 2, 4, 6, or 8
- Divisibility by 3: sum of digits is divisible by 3
- Divisibility by 4: last two digits form a number divisible by 4
- Divisibility by 5: number ends in 0 or 5
- Divisibility by 6: divisible by both 2 and 3
- Divisibility by 8: last three digits form a number divisible by 8
- Divisibility by 9: sum of digits is divisible by 9
- Divisibility by 10: number ends in 0
Definitions
Divisibility Rule
A shortcut method to determine if one number divides another without remainder
Importance
Saves time in factorization and simplification problems
Common mistakes
- Confusing divisibility by 6 (need both 2 AND 3) with divisibility by either 2 OR 3
- Not continuing digit sum process for divisibility by 3 or 9
- Forgetting that divisibility by 4 uses last TWO digits, by 8 uses last THREE digits
Order of Operations (PEMDAS)
Formulas
PEMDAS Hierarchy: () → ^ → ×÷ → +-
Example
2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50
Variables
() = parentheses, ^ = exponents, ×÷ = multiply/divide, +- = add/subtract
Application
Determine correct order for evaluating expressions
Exam tips
- Write out each step clearly to avoid errors
- Use the phrase 'Please Excuse My Dear Aunt Sally' to remember PEMDAS
- When in doubt, add parentheses to clarify your intended order
Key points
- PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)
- Operations of equal precedence are performed left to right
- Always work inside parentheses first, including nested parentheses
- Exponents are calculated before multiplication and division
- Multiplication and division have equal precedence
- Addition and subtraction have equal precedence
Definitions
Order of Operations
The agreed-upon sequence for performing mathematical operations in expressions
Importance
Ensures everyone gets the same answer for the same expression
Common mistakes
- Performing operations left to right without considering precedence
- Forgetting to calculate exponents before multiplication
- Not working inside parentheses completely before moving to next operation
Properties of Real Numbers
Formulas
Distributive Property: a(b + c) = ab + ac
Example
3(x + 4) = 3x + 12
Variables
a = multiplier, b and c = terms being added
Application
Simplify expressions and factor polynomials
Exam tips
- Use properties strategically to make calculations easier
- Recognize when to apply distributive property in reverse (factoring)
- Properties are tools - practice using them in different contexts
Key points
- Commutative Property: a + b = b + a and a × b = b × a
- Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
- Distributive Property: a(b + c) = ab + ac
- Identity Properties: a + 0 = a and a × 1 = a
- Inverse Properties: a + (-a) = 0 and a × (1/a) = 1
- These properties help simplify expressions and solve equations efficiently
Definitions
Commutative Property
Order of numbers doesn't affect the result in addition and multiplication
Importance
Allows flexible arrangement of terms for easier calculation
Distributive Property
Multiplication distributes over addition and subtraction
Importance
Essential for expanding expressions and factoring
Common mistakes
- Thinking subtraction and division are commutative (they're not: 5-3 ≠ 3-5)
- Forgetting to distribute to ALL terms inside parentheses
- Confusing additive identity (0) with multiplicative identity (1)
Fractions and Decimals
Formulas
a/b + c/d = (ad + bc)/(bd)
Example
1/3 + 1/4 = (4 + 3)/12 = 7/12
Variables
a, b, c, d = integers, b ≠ 0, d ≠ 0
Application
Add fractions with different denominators
a/b × c/d = (ac)/(bd)
Example
2/3 × 3/4 = 6/12 = 1/2
Variables
a, b, c, d = integers, b ≠ 0, d ≠ 0
Application
Multiply fractions
a/b ÷ c/d = a/b × d/c = (ad)/(bc)
Example
2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3
Variables
a, b, c, d = integers, b ≠ 0, c ≠ 0, d ≠ 0
Application
Divide fractions
Exam tips
- Always simplify fractions to lowest terms using GCF
- Convert mixed numbers to improper fractions before operations
- Check if answer makes sense in context of the problem
- Practice converting between fractions, decimals, and percentages
Key points
- Fractions represent parts of a whole: numerator/denominator
- Equivalent fractions: multiply/divide numerator and denominator by same number
- To add/subtract fractions: find common denominator
- To multiply fractions: multiply numerators, multiply denominators
- To divide fractions: multiply by reciprocal
- Convert between fractions and decimals using division or multiplication
Definitions
Proper Fraction
Fraction where numerator is less than denominator
Importance
Represents values less than 1
Improper Fraction
Fraction where numerator is greater than or equal to denominator
Importance
Can be converted to mixed numbers
Mixed Number
Combination of whole number and proper fraction
Importance
Often easier to visualize and use in real-world problems
Common mistakes
- Adding denominators when adding fractions (only add numerators)
- Not finding common denominator before adding/subtracting
- Forgetting to flip the second fraction when dividing
- Not simplifying final answers to lowest terms
Connections
- Prime factorization connects to LCM/GCF calculations and fraction simplification
- Divisibility rules speed up factor finding and fraction operations
- PEMDAS applies to all arithmetic operations including fractions
- Properties of real numbers apply to fractions and help in algebraic manipulation
- These concepts form the foundation for algebra, geometry, and advanced mathematics
Quick review questions
Find the LCM and GCF of 24 and 36
Answer
LCM = 72, GCF = 12
Prime factorization: 24 = 2³×3, 36 = 2²×3². LCM = 2³×3² = 72, GCF = 2²×3¹ = 12. Check: 72×12 = 864 = 24×36 ✓
Is 2,847 divisible by 9?
Answer
No
Sum of digits: 2+8+4+7 = 21, then 2+1 = 3. Since 3 is not divisible by 9, 2,847 is not divisible by 9.
Evaluate: 2 + 3 × 4² - (5 + 1)
Answer
44
Step 1: (5+1) = 6. Step 2: 4² = 16. Step 3: 3×16 = 48. Step 4: 2+48-6 = 44
Simplify: 3/4 + 2/3
Answer
17/12
Common denominator is 12. 3/4 = 9/12, 2/3 = 8/12. So 9/12 + 8/12 = 17/12
What is 2/3 ÷ 4/5?
Answer
5/6
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Diagrams
Arithmetic Concepts Overview
Complete overview of all arithmetic concepts covered in this chapter, showing relationships between different topics
Problem-Solving Decision Tree for Arithmetic
Decision flowchart to help students choose the right approach for different types of arithmetic problems
Prime Factorization Process
Step-by-step process for finding prime factorization of any composite number
PEMDAS Order of Operations
Visual guide showing the correct sequence for evaluating mathematical expressions using PEMDAS
Exam strategy
Focus on memorizing divisibility rules and prime numbers up to 50. Practice prime factorization until it becomes automatic. Master LCM/GCF using both listing and prime factorization methods. Always check your work using the relationship LCM × GCF = product of numbers. For PEMDAS problems, write out each step clearly. When working with fractions, always simplify to lowest terms and convert mixed numbers to improper fractions before operating. Use the properties of real numbers to make calculations easier and faster.