1. Stating the problem: Find the Greatest Common Divisor (GCD) for each given set of numbers: (a) 12 and 42, (b) 65 and 91, (c) 12, 16, and 20, (d) 25, 70, and 100. 2. For part (a): - Find prime factors of 12: $$12 = 2^2 \times 3$$ - Find prime factors of 42: $$42 = 2 \times 3 \times 7$$ - Common prime factors: 2 and 3 - GCD is product of common factors: $$2 \times 3 = 6$$ 3. For part (b): - Prime factors of 65: $$65 = 5 \times 13$$ - Prime factors of 91: $$91 = 7 \times 13$$ - Common prime factor: 13 - GCD is 13 4. For part (c): - Prime factors of 12: $$12 = 2^2 \times 3$$ - Prime factors of 16: $$16 = 2^4$$ - Prime factors of 20: $$20 = 2^2 \times 5$$ - Common prime factor: 2 (lowest power is $2^2$) - GCD is $$2^2 = 4$$ 5. For part (d): - Prime factors of 25: $$25 = 5^2$$ - Prime factors of 70: $$70 = 2 \times 5 \times 7$$ - Prime factors of 100: $$100 = 2^2 \times 5^2$$ - Common prime factor: 5 (lowest power is $5^1$ since 70 has only one 5) - GCD is 5 Final answers: (a) GCD = 6 (b) GCD = 13 (c) GCD = 4 (d) GCD = 5