**Prime Time Worksheet Solutions** 1. **Section A: Circle the prime numbers** Prime numbers are numbers greater than 1 that have only two factors: 1 and itself. Numbers listed: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 27 30 33 43 49 81 89 Prime numbers circled: 2, 3, 5, 7, 11, 13, 17, 19, 43, 89 2. **Section B: Prime numbers between 20, 30, and 40** Between 20 and 30: 23, 29 Between 30 and 40: 31, 37 3. **Fill in the blanks:** 1. A number which has only two factors, 1 and itself, is called a **prime** number. 2. The smallest perfect number is **6**. 3. The HCF of two prime numbers is always **1** (because they have no common factors besides 1). 4. The LCM of two co-prime numbers is equal to their **product**. 5. The product of HCF and LCM of two numbers = **product** of the two numbers. 4. **Factors and Multiples** (a) Factors of 18: 1, 2, 3, 6, 9, 18 (b) Factors of 242: 1, 2, 11, 22, 121, 242 First five multiples: (a) 7: 7, 14, 21, 28, 35 (b) 92: 92, 184, 276, 368, 460 5. **HCF by Prime Factorization:** (a) 18 = $2 \times 3^2$ 24 = $2^3 \times 3$ Common prime factors: $2^1$ and $3^1$ HCF = $2 \times 3 = 6$ (b) 65 = $5 \times 13$ 91 = $7 \times 13$ Common prime factor: 13 HCF = 13 6. **LCM by Prime Factorization:** Numbers: 32, 68, 95 32 = $2^5$ 68 = $2^2 \times 17$ 95 = $5 \times 19$ LCM = $2^5 \times 5 \times 17 \times 19 = 31680$ 7. **HCF and LCM using Common Division Method:** Numbers: 96, 120, 180 HCF = 12 LCM = 720 8. **Verify HCF and LCM relationship for pairs:** (a) 95 and 120 HCF = 5, LCM = 2280 Verify: $5 \times 2280 = 11400$ and $95 \times 120 = 11400$ ✓ (b) 15 and 20 HCF = 5, LCM = 60 Verify: $5 \times 60 = 300$ and $15 \times 20 = 300$ ✓ 9. **Prime Factorization of given numbers:** - 88 = $2 \times 2 \times 2 \times 11$ - 90 = $2 \times 3 \times 3 \times 5$ - 126 = $2 \times 3 \times 3 \times 7$ - 75 = $3 \times 5 \times 5$ - 84 = $2 \times 2 \times 3 \times 7$ - 80 = $2 \times 2 \times 2 \times 2 \times 5$ 10. **Word Problems:** (1) Smallest number divisible by 10, 12, 18 is LCM(10,12,18) Prime factors: 10 = $2 \times 5$ 12 = $2^2 \times 3$ 18 = $2 \times 3^2$ LCM = $2^2 \times 3^2 \times 5 = 180$ (2) Greatest number dividing 45 and 75 exactly is HCF(45,75) 45 = $3^2 \times 5$ 75 = $3 \times 5^2$ HCF = $3 \times 5 = 15$ (3) Bells ring every 10, 15, 20 minutes. They start together at 9 a.m. LCM(10,15,20) = $2^2 \times 3 \times 5 = 60$ minutes They ring together again after 60 minutes at 10:00 a.m. (4) Greatest length of rope cut from 36 m and 60 m without remainder is HCF(36,60) 36 = $2^2 \times 3^2$ 60 = $2^2 \times 3 \times 5$ HCF = $2^2 \times 3 = 12$ meters (5) Is 496 a perfect number? A perfect number equals the sum of its proper divisors. Divisors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248 Sum = 496 So yes, 496 is a perfect number. **Final Answers Summary:** - Prime numbers circled: 2,3,5,7,11,13,17,19,43,89 - Prime numbers between 20 & 30: 23,29 - Prime numbers between 30 & 40: 31,37 - Blanks: prime, 6, 1, product, product - Factors of 18: 1,2,3,6,9,18 - Factors of 242: 1,2,11,22,121,242 - Multiples of 7: 7,14,21,28,35 - Multiples of 92: 92,184,276,368,460 - HCF(18,24): 6 - HCF(65,91): 13 - LCM(32,68,95): 31680 - HCF(96,120,180): 12 - LCM(96,120,180): 720 - HCF and LCM verify for pairs (95,120) and (15,20) - Prime factorizations given - Word problems answers: 180, 15, 60 min, 12 m, yes