Whole Number Concepts: – Factors, multiples, primes, squares, cubes.

WHOLE NUMBER CONCEPTS Factors, multiples, primes, squares, cubes. Grade 7 Mathematics: Whole Number Concepts – Factors, Multiples, Primes, Squares, Cubes Subtopic Navigator Introduction Factors Multiples Prime Numbers Squares and Cubes Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Define and identify factors and multiples of whole numbers. Explain prime numbers, composite numbers, and prime factorization. Evaluate perfect squares and cubes, and solve related problems. Apply these concepts in solving intermediate and challenging number problems. Lesson Introduction Whole numbers are the building blocks of arithmetic. Understanding their properties such as factors, multiples, primes, squares, and cubes equips us with skills for higher mathematics. These concepts are crucial in solving divisibility, number theory, and algebra-related problems. Factors A factor of a number is a whole number that divides the number exactly without leaving a remainder. For example, factors of [latex]24[/latex] are [latex]1, 2, 3, 4, 6, 8, 12, 24[/latex]. Example 1: Find all factors of [latex]72[/latex]. Solution: Start by testing divisibility: [latex]72 div 1 = 72, quad 72 div 2 = 36, quad 72 div 3 = 24, quad 72 div 4 = 18, quad 72 div 6 = 12, quad 72 div 8 = 9[/latex]. Hence, factors of [latex]72[/latex] are [latex]1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72[/latex]. Example 2: Find the greatest common factor (GCF) of [latex]108[/latex] and [latex]180[/latex]. Solution: Prime factorization: [latex]108 = 2^2 times 3^3, quad 180 = 2^2 times 3^2 times 5[/latex]. Common factors: [latex]2^2 times 3^2 = 36[/latex]. GCF = [latex]36[/latex]. Example 3: Find the least common multiple (LCM) of [latex]45[/latex] and [latex]120[/latex]. Solution: Prime factorization: [latex]45 = 3^2 times 5, quad 120 = 2^3 times 3 times 5[/latex]. LCM = [latex]2^3 times 3^2 times 5 = 360[/latex]. Exercises (Factors) Find the factors of [latex]96[/latex]. Determine the GCF of [latex]84[/latex] and [latex]132[/latex]. Multiples A multiple of a number is obtained by multiplying it with whole numbers. For example, multiples of [latex]6[/latex] are [latex]6, 12, 18, 24, ldots[/latex]. Example 4: Find the first 8 multiples of [latex]15[/latex]. Solution: [latex]15, 30, 45, 60, 75, 90, 105, 120[/latex]. Example 5: Find the smallest common multiple of [latex]28[/latex] and [latex]42[/latex]. Solution: [latex]28 = 2^2 times 7, quad 42 = 2 times 3 times 7[/latex]. LCM = [latex]2^2 times 3 times 7 = 84[/latex]. Example 6: If a number is a multiple of both [latex]8[/latex] and [latex]12[/latex], what is the smallest such number? Solution: LCM of [latex]8[/latex] and [latex]12[/latex] is [latex]24[/latex]. Hence, the smallest number is [latex]24[/latex]. Exercises (Multiples) Find the first 10 multiples of [latex]18[/latex]. Determine the LCM of [latex]54[/latex] and [latex]90[/latex]. Prime Numbers A prime number has exactly two factors: [latex]1[/latex] and itself. Examples: [latex]2, 3, 5, 7, 11[/latex]. Composite numbers have more than two factors. Prime factorization breaks a number into prime factors. Example 7: Express [latex]315[/latex] as a product of prime factors. Solution: [latex]315 div 3 = 105, quad 105 div 3 = 35, quad 35 div 5 = 7, quad 7 div 7 = 1[/latex]. Hence, [latex]315 = 3^2 times 5 times 7[/latex]. Example 8: Find the HCF of [latex]252[/latex] and [latex]378[/latex] using prime factors. Solution: [latex]252 = 2^2 times 3^2 times 7, quad 378 = 2 times 3^3 times 7[/latex]. HCF = [latex]2 times 3^2 times 7 = 126[/latex]. Example 9: Find the smallest prime greater than [latex]100[/latex]. Solution: Test: [latex]101[/latex] is not divisible by primes ≤ 10. Thus, [latex]101[/latex] is prime. Exercises (Prime Numbers) Write [latex]720[/latex] as a product of prime factors. Find the largest prime factor of [latex]693[/latex]. Squares and Cubes A square number is obtained by multiplying a number by itself: [latex]n^2[/latex]. A cube number is obtained by multiplying a number three times: [latex]n^3[/latex]. These numbers play a vital role in algebra, geometry, and higher math. Example 10: Find the square of [latex]48[/latex]. Solution: [latex]48^2 = 2304[/latex]. Example 11: Find the cube of [latex]25[/latex]. Solution: [latex]25^3 = 15625[/latex]. Example 12: Is [latex]729[/latex] a perfect square or cube? Solution: [latex]sqrt{729} = 27[/latex], so it is a perfect square. [latex]sqrt[3]{729} = 9[/latex], so it is also a perfect cube. Exercises (Squares and Cubes) Evaluate [latex]65^2[/latex]. Find the cube root of [latex]2197[/latex]. Applications and Mixed Problems Example 13: Three bells ring at intervals of 18, 24, and 30 minutes. If they all ring at 9:00 AM, when will they next ring together? Solution: LCM of [latex]18, 24, 30 = 360[/latex] minutes = 6 hours. Next ring together at 3:00 PM. Example 14: A hall has 72 chairs and 108 tables. Find the greatest number of groups that can be formed if each group has equal chairs and equal tables. Solution: GCF of [latex]72[/latex] and [latex]108[/latex] is [latex]36[/latex]. Hence, 36 groups. Example 15: A square park has area [latex]4900 , m^2[/latex]. Find its side and diagonal length. Solution: Side = [latex]sqrt{4900} = 70[/latex] m. Diagonal = [latex]70sqrt{2}[/latex] ≈ 99 m. Exercises (Applications) Two tankers contain 504 litres and 936 litres of petrol. What is the largest capacity of a container that can measure both exactly? A farmer plants trees in rows of 18, 24, and 30. What is the least number of trees he must plant so each row has the same number? Cumulative Exercises Find all factors of [latex]144[/latex]. Write [latex]840[/latex] as a product of prime factors. Determine the LCM of [latex]120[/latex] and [latex]150[/latex]. Evaluate [latex]34^2[/latex] and [latex]21^3[/latex]. Find the largest prime factor of [latex]1287[/latex]. A rectangular hall measures 96 m by 120 m. Find the largest square tile that can cover the floor without cutting. Is [latex]4913[/latex] a perfect cube? Find the GCF of [latex]288[/latex] and [latex]252[/latex]. The product of two primes is [latex]221[/latex]. Find the primes. A circular garden has area [latex]1764 , m^2[/latex]. Find its radius if [latex]pi = 3.142[/latex]. Conclusion/Recap In this lesson, we studied whole number concepts: factors, multiples, prime numbers, squares, and cubes. These ideas strengthen problem-solving in algebra, geometry, and advanced arithmetic. Mastery of these ensures readiness for higher-level mathematics and competitive exams. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c