1. **Stating the Problem:** Find the GCF of 24 and 36 using the listing factors method. 2. **Prime Factorization:** - 24 = $2^3 \times 3^1$ - 36 = $2^2 \times 3^2$ 3. **Listing Factors:** - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 4. **Finding GCF:** Common factors are 1, 2, 3, 4, 6, 12. Greatest is 12. --- 1. **Problem:** GCF of 40 and 60. 2. **Prime Factorization:** - 40 = $2^3 \times 5^1$ - 60 = $2^2 \times 3^1 \times 5^1$ 3. **Common factors:** $2^2 \times 5^1 = 20$ --- 1. **Problem:** GCF of 42 and 56. 2. **Prime Factorization:** - 42 = $2^1 \times 3^1 \times 7^1$ - 56 = $2^3 \times 7^1$ 3. **Common factors:** $2^1 \times 7^1 = 14$ --- 1. **Problem:** GCF of 81 and 63. 2. **Prime Factorization:** - 81 = $3^4$ - 63 = $3^2 \times 7^1$ 3. **Common factors:** $3^2 = 9$ --- 1. **Problem:** GCF of 48 and 60. 2. **Prime Factorization:** - 48 = $2^4 \times 3^1$ - 60 = $2^2 \times 3^1 \times 5^1$ 3. **Common factors:** $2^2 \times 3^1 = 12$ --- 1. **Problem:** GCF of 84 and 126. 2. **Prime Factorization:** - 84 = $2^2 \times 3^1 \times 7^1$ - 126 = $2^1 \times 3^2 \times 7^1$ 3. **Common factors:** $2^1 \times 3^1 \times 7^1 = 42$ --- 1. **Problem:** GCF of 150 and 210. 2. **Prime Factorization:** - 150 = $2^1 \times 3^1 \times 5^2$ - 210 = $2^1 \times 3^1 \times 5^1 \times 7^1$ 3. **Common factors:** $2^1 \times 3^1 \times 5^1 = 30$ --- 1. **Problem:** GCF of 64 and 96. 2. **Prime Factorization:** - 64 = $2^6$ - 96 = $2^5 \times 3^1$ 3. **Common factors:** $2^5 = 32$ --- 1. **Problem:** Find gcd(252, 198) using Euclidean Algorithm. 2. **Steps:** - 252 mod 198 = 54 - 198 mod 54 = 36 - 54 mod 36 = 18 - 36 mod 18 = 0 3. **Result:** gcd = 18 --- 1. **Problem:** Find gcd(420, 308). 2. **Steps:** - 420 mod 308 = 112 - 308 mod 112 = 84 - 112 mod 84 = 28 - 84 mod 28 = 0 3. **Result:** gcd = 28 --- 1. **Problem:** Find gcd(315, 210). 2. **Steps:** - 315 mod 210 = 105 - 210 mod 105 = 0 3. **Result:** gcd = 105 --- 1. **Problem:** Find $x$ and $y$ for $a=81$, $b=57$ using Extended Euclidean Algorithm. 2. **Steps:** - gcd(81,57) steps: - 81 = 57*1 + 24 - 57 = 24*2 + 9 - 24 = 9*2 + 6 - 9 = 6*1 + 3 - 6 = 3*2 + 0 - gcd = 3 3. Back substitution: - 3 = 9 - 6*1 - 6 = 24 - 9*2 - 3 = 9 - (24 - 9*2)*1 = 3*9 - 1*24 - 9 = 57 - 24*2 - 3 = 3*(57 - 24*2) - 1*24 = 3*57 - 7*24 - 24 = 81 - 57*1 - 3 = 3*57 - 7*(81 - 57*1) = -7*81 + 10*57 4. **Solution:** $x = -7$, $y = 10$, satisfying $81x + 57y = 3$ --- 1. **Problem:** Find GCD of 120 and 35 and express as $120x + 35y = \text{GCD}$. 2. **Euclidean Algorithm:** - 120 mod 35 = 15 - 35 mod 15 = 5 - 15 mod 5 = 0 - gcd = 5 3. **Extended Algorithm back substitution:** - 5 = 35 - 15*2 - 15 = 120 - 35*3 - 5 = 35 - (120 - 35*3)*2 = 35*7 - 120*2 4. **Answer:** $x = -2$, $y = 7$ so that $120(-2) + 35(7) = 5$ --- 1. **Problem:** Find whole numbers $x, y$ such that $39x + 14y = 1$ with $x > 5$. 2. **Euclidean Algorithm:** - 39 mod 14 = 11 - 14 mod 11 = 3 - 11 mod 3 = 2 - 3 mod 2 = 1 - 2 mod 1 = 0 - gcd = 1 3. **Back substitution:** - 1 = 3 - 2*1 - 2 = 11 - 3*3 - 1 = 3 - (11 - 3*3)*1 = 4*3 - 11 - 3 = 14 - 11*1 - 1 = 4*(14 - 11) - 11 = 4*14 - 5*11 - 11 = 39 - 14*2 - 1 = 4*14 - 5*(39 - 14*2) = 14*14 - 5*39 4. Current solution: $x = -5$, $y = 14$, but $x = -5 \not> 5$. 5. **General solution:** Add multiples of $(14, -39)$: - $x = -5 + 14k$, $y = 14 - 39k$ - For $x > 5$, choose $k=1$: - $x = 9$, $y = 14 - 39 = -25$ 6. **Final answer:** $x=9$, $y=-25$ satisfy $39x + 14y = 1$ with $x > 5$. **Final answers:** 1) 12 2) 20 3) 14 4) 9 5) 12 6) 42 7) 30 8) 32 9) 18 10) 28 11) 105 12) $x=-7$, $y=10$ 13) gcd=5, $x=-2$, $y=7$ 14) $x=9$, $y=-25$