Arithmetic — Multiples, Factors, PEMDAS, Fractions & Decimals - Slides
This chapter covers fundamental arithmetic concepts essential for UPCAT and other college entrance exams. You will learn about multiples and factors, prime factorization, LCM and GCF calculations, divisibility rules, order of operations (PEMDAS), and operations with fractions and decimals. These skills form the foundation for advanced mathematical problem-solving.
Slides
Understanding Multiples and Factors
Notes
Remember: Every integer has 1 as its smallest factor and itself as its largest factor. The smallest multiple of any integer is the integer itself.
Examples
- 24 has factors: 1, 2, 3, 4, 6, 8, 12, 24
- Multiples of 6: 6, 12, 18, 24, 30, 36...
- Since 6 is a factor of 24, then 24 is a multiple of 6
Understanding the relationship between multiples and factors is crucial for solving arithmetic problems. Think of multiples as the 'counting by' pattern (5, 10, 15, 20...) and factors as the numbers that divide evenly into another number.
Main points
- Multiple: A positive integer m is a multiple of n if m divides n without remainder
- Factor: A positive integer m is a factor of n if n divides m without remainder
- Key relationship: If m is a factor of n, then n is a multiple of m
- Example: 25 is a multiple of 5 because 25 ÷ 5 = 5 (no remainder)
Diagram
Decision tree for determining factor and multiple relationships between two numbers
Prime and Composite Numbers
Notes
To check if a large number n is prime, only test divisibility by primes up to √n. This saves time and effort in calculations.
Examples
- Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
- Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16...
- Is 17 prime? Check factors: only 1 and 17, so YES
- Is 21 prime? Factors: 1, 3, 7, 21, so NO (composite)
Prime numbers are the building blocks of all integers. Understanding them helps in factorization and solving complex arithmetic problems. Composite numbers can be broken down into prime factors.
Main points
- Prime number: Has exactly two factors - 1 and itself
- Composite number: Has more than two factors
- Special cases: 0 and 1 are neither prime nor composite
- 2 is the smallest and only even prime number
Diagram
Classification of numbers into prime, composite, and special cases
Testing for Prime Numbers - Step by Step
Notes
This method is much faster than testing all numbers. For example, to test if 100 is prime, you only need to check divisibility by 2, 3, 5, 7 instead of all numbers up to 100.
Examples
- Test 431: √431 ≈ 20.8, so check primes 2, 3, 5, 7, 11, 13, 17, 19
- 431 ÷ 2 = 215.5 (not divisible)
- 431 ÷ 3 = 143.67 (not divisible)
- Continue testing... None divide evenly, so 431 is prime
This systematic approach saves time when testing large numbers for primality. You only need to check prime divisors up to the square root because any factor larger than √n would have a corresponding factor smaller than √n.
Main points
- Step 1: Find √n (square root of the number)
- Step 2: Test divisibility by all prime numbers ≤ √n
- Step 3: If any prime divides n, then n is composite
- Step 4: If no prime divides n, then n is prime
Diagram
Algorithm for testing if a number is prime or composite
Prime Factorization Using Factor Trees
Notes
The choice of initial factors doesn't matter - you'll always get the same prime factorization. This is called the Fundamental Theorem of Arithmetic.
Examples
- Factor 100: 100 = 4 × 25 = (2 × 2) × (5 × 5) = 2² × 5²
- Factor 100: 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2² × 5²
- Factor 72: 72 = 8 × 9 = (2³) × (3²) = 2³ × 3²
Prime factorization breaks down any composite number into its prime building blocks. The factor tree method provides a visual way to systematically find all prime factors. No matter which factors you choose first, you'll always get the same final result.
Main points
- Express number as product of two smaller numbers
- Draw branches from number to its factors
- Continue factoring composite numbers
- Stop when all leaves are prime numbers
Diagram
Factor tree showing the prime factorization of 100
Least Common Multiple (LCM) - Two Methods
Notes
For large numbers, the prime factorization method is much more efficient. Remember: LCM × GCF = product of the two original numbers.
Examples
- Find LCM of 12 and 18:
- Method 1: Multiples of 12: 12, 24, 36, 48... Multiples of 18: 18, 36, 54... LCM = 36
- Method 2: 12 = 2² × 3, 18 = 2 × 3². Take highest powers: 2² × 3² = 4 × 9 = 36
The LCM is essential for fraction operations and word problems involving repeated events. The prime factorization method is more efficient for larger numbers, while listing multiples works well for smaller numbers.
Main points
- Method 1: List multiples until you find the smallest common one
- Method 2: Use prime factorization with highest powers
- LCM is the smallest number divisible by both given numbers
- Useful for adding fractions with different denominators
Diagram
Two methods for finding the Least Common Multiple of two numbers
Greatest Common Factor (GCF) - Finding the Largest Common Divisor
Notes
When one number is a factor of another, the smaller number is automatically the GCF. For example, GCF of 12 and 36 is 12.
Examples
- Find GCF of 24 and 36:
- Method 1: Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36. GCF = 12
- Method 2: 24 = 2³ × 3, 36 = 2² × 3². Take lowest powers: 2² × 3¹ = 4 × 3 = 12
The GCF helps simplify fractions and solve problems involving equal groups. Understanding both methods gives you flexibility in choosing the most efficient approach based on the numbers involved.
Main points
- Method 1: List all factors and find the largest common one
- Method 2: Use prime factorization with lowest powers
- GCF is the largest number that divides both given numbers
- Useful for simplifying fractions to lowest terms
Diagram
Two methods for finding the Greatest Common Factor of two numbers
Divisibility Rules - Quick Tests for Common Divisors
Notes
For divisibility by 6, a number must satisfy both the rule for 2 AND the rule for 3. For larger numbers, you can repeat the digit sum process until you get a single digit.
Examples
- Is 1,236 divisible by 3? 1+2+3+6 = 12, and 1+2 = 3. Yes!
- Is 1,235 divisible by 5? Ends in 5. Yes!
- Is 1,234 divisible by 2? Ends in 4. Yes!
- Is 1,237 divisible by 9? 1+2+3+7 = 13, 1+3 = 4. No!
Divisibility rules help you quickly determine if one number divides another without performing actual division. These shortcuts are invaluable for factorization, simplifying fractions, and solving arithmetic problems efficiently.
Main points
- Divisible by 2: Number ends in 0, 2, 4, 6, or 8
- Divisible by 3: Sum of digits is divisible by 3
- Divisible by 5: Number ends in 0 or 5
- Divisible by 9: Sum of digits is divisible by 9
Diagram
Overview of common divisibility rules for quick mental math
PEMDAS - Order of Operations
Notes
Common mistake: Students often do multiplication before division or addition before subtraction. Remember that operations of equal priority are done from left to right!
Examples
- Evaluate: 3 + 2 × 4² - (8 ÷ 2)
- Step 1: Parentheses: 8 ÷ 2 = 4 → 3 + 2 × 4² - 4
- Step 2: Exponents: 4² = 16 → 3 + 2 × 16 - 4
- Step 3: Multiplication: 2 × 16 = 32 → 3 + 32 - 4
- Step 4: Addition/Subtraction (left to right): 3 + 32 - 4 = 35 - 4 = 31
PEMDAS ensures everyone gets the same answer when evaluating mathematical expressions. Without a standard order, the same expression could have multiple interpretations. Remember that multiplication/division have equal priority, as do addition/subtraction.
Main points
- P - Parentheses (grouping symbols first)
- E - Exponents (powers and roots)
- M - Multiplication and D - Division (left to right)
- A - Addition and S - Subtraction (left to right)
Diagram
Step-by-step process for applying the order of operations (PEMDAS)
PEMDAS Practice - Complex Expression
Notes
When dealing with brackets and parentheses, work from the innermost grouping symbols outward. Always recalculate the expression after each step to avoid carrying forward errors.
Examples
- Evaluate: 2[3 + 4(5² - 3 × 7)] - 8 ÷ 2²
- Step 1: Inner parentheses: 5² = 25, 3 × 7 = 21 → 25 - 21 = 4
- Step 2: 2[3 + 4(4)] - 8 ÷ 2² → 2[3 + 16] - 8 ÷ 4
- Step 3: 2[19] - 8 ÷ 4 → 38 - 2
- Step 4: Final answer = 36
Complex expressions require careful attention to detail and systematic application of PEMDAS. Breaking down the problem into smaller steps helps prevent mistakes and makes the solution clearer.
Main points
- Work through nested parentheses from inside out
- Handle multiple exponents separately
- Be careful with negative numbers and signs
- Double-check each step to avoid errors
Diagram
Step-by-step solution of a complex PEMDAS expression
Understanding Fractions - Parts of a Whole
Notes
To convert mixed numbers to improper fractions: multiply whole number by denominator, add numerator, keep same denominator. Example: 2 3/5 = (2×5+3)/5 = 13/5
Examples
- 3/4 means 3 out of 4 equal parts
- 7/8 is a proper fraction (7 < 8)
- 5/3 is an improper fraction (5 > 3)
- Mixed number: 1 2/3 = 5/3 as improper fraction
Fractions represent parts of a whole or ratios between quantities. Understanding the relationship between numerator and denominator is crucial for all fraction operations. Visual representations help solidify these concepts.
Main points
- Fraction = Numerator/Denominator
- Numerator: number of parts we have
- Denominator: total number of equal parts
- Proper fraction: numerator < denominator
Diagram
Visual representation of the fraction 3/4 showing three-quarters shaded
Adding and Subtracting Fractions - Same Denominators
Notes
Always check if your final answer can be simplified by dividing both numerator and denominator by their GCF. For example, 6/12 = 1/2 after dividing by GCF of 6.
Examples
- Same denominators: 3/7 + 2/7 = 5/7
- Different denominators: 1/4 + 1/6
- LCD of 4 and 6 = 12
- 1/4 = 3/12, 1/6 = 2/12
- 3/12 + 2/12 = 5/12
When fractions have the same denominator, you're working with the same-sized pieces, so you can directly add or subtract the number of pieces. With different denominators, you must first convert to equivalent fractions with a common denominator.
Main points
- Same denominators: Add/subtract numerators, keep denominator
- Different denominators: Find LCD (Least Common Denominator)
- Convert fractions to equivalent forms with LCD
- Simplify the final answer to lowest terms
Diagram
Decision process for adding and subtracting fractions
Multiplying and Dividing Fractions
Notes
When canceling, you can cancel a factor in any numerator with the same factor in any denominator. This makes calculations much easier with large numbers.
Examples
- Multiply: 2/3 × 4/5 = 8/15
- Cancel first: 2/3 × 4/5 = (2×4)/(3×5) = 8/15
- Divide: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
- Mixed: 1 1/2 × 2/3 = 3/2 × 2/3 = 6/6 = 1
Fraction multiplication is straightforward - multiply straight across. Division uses the 'keep, change, flip' rule: keep the first fraction, change division to multiplication, flip the second fraction. Canceling common factors early prevents working with large numbers.
Main points
- Multiplication: Multiply numerators together, multiply denominators together
- Division: Multiply by the reciprocal (flip the second fraction)
- Cancel common factors before multiplying to simplify
- Mixed numbers must be converted to improper fractions first
Diagram
Process for multiplying and dividing fractions
Converting Between Fractions and Decimals
Notes
Memorize common fraction-decimal equivalents: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/5=0.2, 2/5=0.4, 3/5=0.6, 4/5=0.8. These appear frequently on exams.
Examples
- 3/4 to decimal: 3 ÷ 4 = 0.75
- 0.625 to fraction: 625/1000 = 5/8 (after simplifying)
- 1/3 = 0.333... (repeating)
- Common conversions: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
Converting between fractions and decimals is essential for comparing numbers and solving real-world problems. Understanding place values helps in accurate conversion. Some fractions produce terminating decimals, while others repeat infinitely.
Main points
- Fraction to decimal: Divide numerator by denominator
- Decimal to fraction: Use place value, then simplify
- Terminating decimals have exact fraction equivalents
- Repeating decimals require special conversion techniques
Diagram
Process for converting between fractions and decimals
Decimal Operations - Addition, Subtraction, Multiplication
Notes
When adding or subtracting decimals, you can add zeros to the right of the decimal point to make alignment easier. For example, 12.3 + 5.67 can be written as 12.30 + 5.67.
Examples
- Addition: 12.34 + 5.678 = 18.018
- Subtraction: 15.6 - 3.45 = 12.15
- Multiplication: 2.3 × 1.4 = 3.22 (1+1=2 decimal places)
- Division: 8.4 ÷ 0.3 = 84 ÷ 3 = 28
Decimal operations follow similar rules to whole number operations with special attention to decimal point placement. Proper alignment and careful counting of decimal places ensures accurate results.
Main points
- Addition/Subtraction: Align decimal points vertically
- Multiplication: Multiply normally, count total decimal places
- Division: Move decimal point to make divisor whole
- Rounding: Use appropriate precision for the context
Diagram
Process for performing operations with decimal numbers
Problem-Solving with Arithmetic Concepts
Notes
Practice identifying keywords: 'total' suggests addition, 'difference' suggests subtraction, 'times' or 'of' suggests multiplication, 'per' or 'each' suggests division.
Examples
- Problem: A recipe calls for 2/3 cup flour. How much flour for 1 1/2 times the recipe?
- Solution: 2/3 × 1 1/2 = 2/3 × 3/2 = 6/6 = 1 cup
- Problem: Find LCM of 12 and 18 to add 5/12 + 7/18
- LCM = 36, so 5/12 + 7/18 = 15/36 + 14/36 = 29/36
Real-world problems often combine multiple arithmetic concepts. Success depends on recognizing which concepts apply and applying them in the correct sequence. Always verify your answer makes sense in the context of the problem.
Main points
- Identify what type of arithmetic concept is needed
- Set up the problem systematically
- Apply appropriate rules and formulas
- Check your answer for reasonableness
Diagram
Problem-solving strategy for arithmetic word problems
In summary
Mastering these arithmetic fundamentals - multiples, factors, PEMDAS, fractions, and decimals - provides the foundation for success in UPCAT and other college entrance exams. Practice these concepts regularly, focusing on accuracy and speed. Remember to check your work and verify that your answers make sense in context. These skills will serve you well not only on exams but also in advanced mathematical coursework and real-world applications.