1. **Evaluate expressions using a 12-hour clock where ⊕ means addition modulo 12 and ⊖ means subtraction modulo 12.** - a. $3 \oplus 5 = (3 + 5) \mod 12 = 8$ - b. $8 \oplus 4 = (8 + 4) \mod 12 = 12 \mod 12 = 0$ - c. $11 \oplus 3 = (11 + 3) \mod 12 = 14 \mod 12 = 2$ - d. $7 \oplus 9 = (7 + 9) \mod 12 = 16 \mod 12 = 4$ - e. $11 \oplus 10 = (11 + 10) \mod 12 = 21 \mod 12 = 9$ - f. $2 \ominus 7 = (2 - 7) \mod 12 = (-5) \mod 12 = 7$ - g. $10 \ominus 11 = (10 - 11) \mod 12 = (-1) \mod 12 = 11$ - h. $4 \ominus 9 = (4 - 9) \mod 12 = (-5) \mod 12 = 7$ - i. $3 \ominus 8 = (3 - 8) \mod 12 = (-5) \mod 12 = 7$ - j. $1 \ominus 4 = (1 - 4) \mod 12 = (-3) \mod 12 = 9$ 2. **Determine whether the congruences are True or False**: - a. $5 \equiv 8 \mod 3$? Since $5 - 8 = -3$ and $-3 \div 3 = -1$ (an integer), TRUE. - b. $5 \equiv 20 \mod 4$? Since $5 - 20 = -15$ and $-15 \div 4 = -3.75$ (not integer), FALSE. - c. $88 \equiv 5 \mod 9$? $88 - 5 = 83$ and $83 \div 9 \approx 9.222$ (not integer), FALSE. - d. $100 \equiv 20 \mod 8$? $100 - 20 = 80$ and $80 \div 8 = 10$ (integer), TRUE. - e. $25 \equiv 85 \mod 12$? $25 - 85 = -60$ and $-60 \div 12 = -5$ (integer), TRUE. 3. **Perform modular arithmetic calculations:** - a. $(9 + 15) \mod 7 = 24 \mod 7 = 3$ - b. $(5 + 22) \mod 8 = 27 \mod 8 = 3$ - c. $(42 + 35) \mod 3 = 77 \mod 3 = 2$ - d. $(25 - 10) \mod 4 = 15 \mod 4 = 3$ - e. $(60 - 32) \mod 9 = 28 \mod 9 = 1$ - f. $(14 \times 18) \mod 5 = 252 \mod 5 = 2$ - g. $(9 \times 15) \mod 8 = 135 \mod 8 = 7$ - h. $(26 \times 11) \mod 15 = 286 \mod 15 = 1$ - i. $(4 \times 22) \mod 3 = 88 \mod 3 = 1$ - j. $(5 \times 12) \mod 4 = 60 \mod 4 = 0$ **Final answers:** A: a)8, b)0, c)2, d)4, e)9, f)7, g)11, h)7, i)7, j)9 B: a)TRUE, b)FALSE, c)FALSE, d)TRUE, e)TRUE C: a)3, b)3, c)2, d)3, e)1, f)2, g)7, h)1, i)1, j)0