1. **Understanding Prime Factorisation:** Prime factorisation means breaking a number into its prime factors. Among the options given, the correct one is D: breaking a number into prime factors. 2. **Prime Factor Trees for Given Numbers:** Let's break each number into prime factors using factor trees. a. 15: $$15 = 3 \times 5$$ b. 20: $$20 = 2 \times 10 = 2 \times 2 \times 5$$ c. 14: $$14 = 2 \times 7$$ d. 8: $$8 = 2 \times 4 = 2 \times 2 \times 2$$ e. 21: $$21 = 3 \times 7$$ f. 22: $$22 = 2 \times 11$$ g. 16: $$16 = 2 \times 8 = 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2$$ h. 27: $$27 = 3 \times 9 = 3 \times 3 \times 3$$ i. 42: $$42 = 2 \times 21 = 2 \times 3 \times 7$$ j. 30: $$30 = 2 \times 15 = 2 \times 3 \times 5$$ 3. **Different Factor Trees for Some Numbers:** Let's find multiple factor trees for these numbers. a. 40 - $$40 = 4 \times 10 = 2 \times 2 \times 2 \times 5$$ - $$40 = 5 \times 8 = 5 \times 2 \times 2 \times 2$$ b. 48 - $$48 = 6 \times 8 = 2 \times 3 \times 2 \times 2 \times 2$$ - $$48 = 12 \times 4 = 2 \times 2 \times 3 \times 2 \times 2$$ c. 60 - $$60 = 6 \times 10 = 2 \times 3 \times 2 \times 5$$ - $$60 = 4 \times 15 = 2 \times 2 \times 3 \times 5$$ d. 24 - $$24 = 4 \times 6 = 2 \times 2 \times 2 \times 3$$ - $$24 = 3 \times 8 = 3 \times 2 \times 2 \times 2$$ e. 50 - $$50 = 5 \times 10 = 5 \times 2 \times 5$$ - $$50 = 25 \times 2 = 5 \times 5 \times 2$$ f. 72 - $$72 = 8 \times 9 = 2 \times 2 \times 2 \times 3 \times 3$$ - $$72 = 6 \times 12 = 2 \times 3 \times 2 \times 2 \times 3$$ 4. **Prime Factors by Division Method:** We divide each number by smallest prime factor repeatedly. a. 84: $$84 \div 2 = 42$$ $$42 \div 2 = 21$$ $$21 \div 3 = 7$$ $$7 \div 7 = 1$$ Prime factors: $$2 \times 2 \times 3 \times 7$$ b. 117: $$117 \div 3 = 39$$ $$39 \div 3 = 13$$ $$13 \div 13 = 1$$ Prime factors: $$3 \times 3 \times 13$$ c. 333: $$333 \div 3 = 111$$ $$111 \div 3 = 37$$ $$37 \div 37 = 1$$ Prime factors: $$3 \times 3 \times 37$$ d. 126: $$126 \div 2 = 63$$ $$63 \div 3 = 21$$ $$21 \div 3 = 7$$ $$7 \div 7 = 1$$ Prime factors: $$2 \times 3 \times 3 \times 7$$ e. 520: $$520 \div 2 = 260$$ $$260 \div 2 = 130$$ $$130 \div 2 = 65$$ $$65 \div 5 = 13$$ $$13 \div 13 = 1$$ Prime factors: $$2 \times 2 \times 2 \times 5 \times 13$$ f. 99: $$99 \div 3 = 33$$ $$33 \div 3 = 11$$ $$11 \div 11 = 1$$ Prime factors: $$3 \times 3 \times 11$$ 5. **Common Factors of 12 and 18:** Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 Common factors, highlighted as circles: 1, 2, 3, 6.