Number Properties: Practice Questions & Explanations

Prime factorization expresses a number as a product of prime numbers. Process: divide by smallest primes repeatedly. 60 ÷ 2 = 30; 30 ÷ 2 = 15; 15 ÷ 3 = 5; 5 ÷ 5 = 1. So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Primes are natural numbers > 1 divisible only by 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37... Composite numbers are non-prime natural numbers > 1; they can be factored into primes. 1 is neither prime nor composite. 2 is the only even prime. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic). Uses of prime factorization: (1) Finding GCF (greatest common factor) — multiply common prime factors with smallest exponents; (2) Finding LCM (least common multiple) — multiply all prime factors with largest exponents; (3) Simplifying fractions and radicals; (4) Number theory problems. Example: GCF of 60 and 84: 60 = 2²·3·5, 84 = 2²·3·7; common factors: 2², 3; GCF = 4·3 = 12.

GCF (greatest common factor) is the largest number that divides both. Method 1 — prime factorization: 24 = 2³ × 3; 36 = 2² × 3². Common prime factors with smallest exponents: 2² (smaller of 2³ and 2²) and 3¹ (smaller of 3¹ and 3²). GCF = 2² × 3 = 4 × 3 = 12. Method 2 — list factors: factors of 24: 1, 2, 3, 4, 6, 8, 12, 24; factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36; common: 1, 2, 3, 4, 6, 12; greatest is 12. LCM (least common multiple) — smallest number divisible by both: use prime factorization with LARGEST exponents: LCM(24, 36) = 2³ × 3² = 8 × 9 = 72. Relationship: GCF × LCM = product of the numbers. Check: 12 × 72 = 864 = 24 × 36 ✓. Applications: reducing fractions (divide by GCF); adding fractions (LCM as common denominator); cycling problems; tile patterns; arrangement problems.

Percent means 'per hundred'; convert to a decimal or fraction and multiply. 30% = 0.30 = 30/100 = 3/10. So 30% of 80 = 0.30 × 80 = 24. Or use fraction: (3/10) × 80 = 240/10 = 24. Percent applications: (1) Tax: total = price × (1 + tax rate); (2) Discount: sale price = original × (1 − discount rate); (3) Tip: tip = bill × tip rate; (4) Percent change: (new − old)/old × 100%; (5) Percent of: percent × whole. Useful percent equivalents to memorize: 10% = 0.10 = 1/10; 20% = 0.20 = 1/5; 25% = 0.25 = 1/4; 33⅓% = 1/3; 50% = 0.5 = 1/2; 66⅔% = 2/3; 75% = 0.75 = 3/4. Quick mental math: 10% of any number is the number with the decimal moved one place left; for 30%, take 10% and multiply by 3. Percent increase: if price rises 20%, multiply by 1.20; if it rises 50%, multiply by 1.50. Percent decrease: multiply by (1 − percentage).

37 is prime — only divisible by 1 and 37. 21 = 3×7; 27 = 3×9; 51 = 3×17.

Multiples of 4: 4, 8, 12... Multiples of 6: 6, 12... First common multiple = 12.

A percent means 'per hundred,' so 30% = 30/100. Reduce by dividing the numerator and denominator by their greatest common factor, 10: 30/100 = 3/10. The fraction in lowest terms is 3/10. The distractor 3/100 omits the reduction step incorrectly (it confuses 30/100 reduction), and 1/3 is a different value (33.3%). To convert a percent to a fraction, write it over 100 and simplify. Here 30% equals 3/10.