USTET Mathematics — Arithmetic — Multiples, Factors, PEMDAS, Fractions & DecimalsStudy Notes

A multiple of a number is the result when that number is multiplied by any whole number. For example, multiples of 5 are: 5, 10, 15, 20, 25, and so on. A factor of a number divides that number exactly without remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. Step-by-Step Method to Find Factors: 1. Start with 1 and the number itself 2. Test each number from 2 up to the square root of the given number 3. If the number divides evenly, both the divisor and quotient are factors 4. List all factors in ascending order Example 1: Find all factors of 24 Step 1: Start with 1 and 24 Step 2: Test numbers from 2 to √24 ≈ 4.9 - 24 ÷ 2 = 12 (both 2 and 12 are factors) - 24 ÷ 3 = 8 (both 3 and 8 are factors) - 24 ÷ 4 = 6 (both 4 and 6 are factors) Step 3: Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

A prime number has exactly two factors: 1 and itself. Numbers with more than two factors are composite. Prime factorization expresses a number as a product of prime factors. Prime Factorization Method (Factor Tree): 1. Start with the given number 2. Find two factors of the number 3. Draw branches to these factors 4. Continue factoring composite numbers 5. Stop when all branches end in prime numbers 6. Multiply all prime factors to verify Example: Prime factorization of 60 Step 1: 60 = 6 × 10 Step 2: 6 = 2 × 3, 10 = 2 × 5 Step 3: All factors (2, 3, 2, 5) are prime Step 4: 60 = 2² × 3 × 5 Verification: 4 × 3 × 5 = 60 ✓ Primality Test for Large Numbers: 1. Find √n (square root) 2. Test divisibility by all primes up to √n 3. If no prime divides n, then n is prime Example: Is 97 prime? Step 1: √97 ≈ 9.8 Step 2: Test primes 2, 3, 5, 7 - 97 ÷ 2 = 48.5 (not divisible) - 97 ÷ 3 = 32.33... (not divisible) - 97 ÷ 5 = 19.4 (not divisible) - 97 ÷ 7 = 13.86... (not divisible) Step 3: 97 is prime

The Least Common Multiple (LCM) is the smallest number divisible by two or more given numbers. The Greatest Common Factor (GCF) is the largest number that divides two or more given numbers. Method 1 - Listing Method: For LCM: List multiples until you find the first common one For GCF: List factors and find the largest common one Example: LCM and GCF of 12 and 18 Multiples of 12: 12, 24, 36, 48, 60, 72... Multiples of 18: 18, 36, 54, 72, 90... LCM = 36 (first common multiple) Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 GCF = 6 (largest common factor) Method 2 - Prime Factorization: For LCM: Use highest power of each prime factor For GCF: Use lowest power of common prime factors Example: LCM and GCF of 24 and 36 24 = 2³ × 3 36 = 2² × 3² LCM = 2³ × 3² = 8 × 9 = 72 GCF = 2² × 3¹ = 4 × 3 = 12 Important Relationship: LCM × GCF = Product of the numbers Verification: 72 × 12 = 864, and 24 × 36 = 864 ✓

PEMDAS is the order in which mathematical operations should be performed: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). In the Philippines, we also use GEMDAS (Grouping symbols, Exponents, Multiplication/Division, Addition/Subtraction). Step-by-Step PEMDAS Process: 1. Solve expressions inside Parentheses (or grouping symbols) first 2. Calculate Exponents (powers and roots) 3. Perform Multiplication and Division from left to right 4. Perform Addition and Subtraction from left to right Example 1: Solve 3 + 4 × 2² Step 1: No parentheses Step 2: Exponents first: 2² = 4, so we have 3 + 4 × 4 Step 3: Multiplication: 4 × 4 = 16, so we have 3 + 16 Step 4: Addition: 3 + 16 = 19 Example 2: Solve (15 - 3) ÷ 4 + 2 × 5 Step 1: Parentheses: (15 - 3) = 12, so we have 12 ÷ 4 + 2 × 5 Step 2: No exponents Step 3: Division and multiplication (left to right): 12 ÷ 4 = 3, then 2 × 5 = 10, so we have 3 + 10 Step 4: Addition: 3 + 10 = 13 Common Mistakes to Avoid: - Don't work left to right ignoring PEMDAS - Remember MD and AS are equal priority (work left to right) - Always do operations inside parentheses first - Don't forget to apply PEMDAS to expressions within parentheses

A fraction represents a part of a whole, written as numerator/denominator. Understanding fraction operations is essential for algebra and real-world problem solving. Adding and Subtracting Fractions: Step 1: Find the LCM of denominators (common denominator) Step 2: Convert each fraction to equivalent fraction with common denominator Step 3: Add or subtract numerators, keep the common denominator Step 4: Simplify by dividing by GCF if possible Example: 3/4 + 5/6 Step 1: LCM of 4 and 6 = 12 Step 2: 3/4 = 9/12, 5/6 = 10/12 Step 3: 9/12 + 10/12 = 19/12 Step 4: 19/12 is already in simplest form Multiplying Fractions: Step 1: Multiply numerators together Step 2: Multiply denominators together Step 3: Simplify the result Example: 2/3 × 4/5 Step 1: 2 × 4 = 8 Step 2: 3 × 5 = 15 Step 3: 8/15 (already simplified) Dividing Fractions: Step 1: Keep the first fraction unchanged Step 2: Change division to multiplication Step 3: Flip the second fraction (find reciprocal) Step 4: Multiply and simplify Example: 3/5 ÷ 2/7 Step 1: Keep 3/5 Step 2: Change ÷ to × Step 3: Flip 2/7 to get 7/2 Step 4: 3/5 × 7/2 = 21/10

Decimals represent fractions with denominators of powers of 10. Converting between fractions and decimals is crucial for calculations and comparisons. Converting Fractions to Decimals: Method: Divide the numerator by the denominator Example 1: Convert 3/4 to decimal 3 ÷ 4 = 0.75 Example 2: Convert 1/3 to decimal 1 ÷ 3 = 0.333... = 0.3̄ (repeating decimal) Converting Decimals to Fractions: Step 1: Count decimal places Step 2: Write decimal over power of 10 Step 3: Simplify using GCF Example: Convert 0.75 to fraction Step 1: 2 decimal places Step 2: 75/100 Step 3: GCF of 75 and 100 is 25, so 75/100 = 3/4 Decimal Operations: Addition/Subtraction: Align decimal points, add/subtract normally Multiplication: Multiply as whole numbers, count total decimal places Division: Use long division or convert to fractions Example: 2.5 × 0.4 Step 1: 25 × 4 = 100 Step 2: Count decimal places: 1 + 1 = 2 Step 3: 100 with 2 decimal places = 1.00 = 1 Percentages: To convert: Decimal × 100 = Percentage Example: 0.65 = 65%, 1.25 = 125%

Divisibility rules help determine if one number divides another without performing actual division. These rules are time-savers in exams and useful for factorization. Key Divisibility Rules: Divisible by 2: Last digit is even (0, 2, 4, 6, 8) Example: 1,234 is divisible by 2 (last digit is 4) Divisible by 3: Sum of digits is divisible by 3 Example: 342 → 3+4+2 = 9, and 9÷3 = 3, so 342 is divisible by 3 Divisible by 4: Last two digits form a number divisible by 4 Example: 1,316 → 16÷4 = 4, so 1,316 is divisible by 4 Divisible by 5: Last digit is 0 or 5 Example: 745 and 1,250 are both divisible by 5 Divisible by 6: Must be divisible by both 2 and 3 Example: 48 is even and 4+8=12 is divisible by 3, so 48 is divisible by 6 Divisible by 9: Sum of digits is divisible by 9 Example: 729 → 7+2+9 = 18, and 18÷9 = 2, so 729 is divisible by 9 Divisible by 10: Last digit is 0 Example: 1,570 is divisible by 10 Practical Application: To check if 2,346 is divisible by 6: Step 1: Check divisibility by 2: Last digit is 6 (even) ✓ Step 2: Check divisibility by 3: 2+3+4+6 = 15, 15÷3 = 5 ✓ Step 3: Since divisible by both 2 and 3, it's divisible by 6 Verification: 2,346 ÷ 6 = 391