1. Problem 33: Find the number of distinct incongruent solutions to $$98x \equiv 175 \pmod{49}$$. Step 1: Simplify the congruence. Note that $49$ divides both sides since modulo $49$, we reduce coefficients: $$98 \equiv 0 \pmod{49}\quad \text{because } 98 = 2 \times 49$$ $$175 \equiv 175 - 3 \times 49 = 175 - 147 = 28 \pmod{49}$$ Thus the congruence becomes: $$0 \cdot x \equiv 28 \pmod{49} \Rightarrow 0 \equiv 28 \pmod{49}$$ Step 2: Since $0 \neq 28 \pmod{49}$, there are no solutions unless $28 \equiv 0$ mod $49$, which it does not. However, re-checking the original problem: The number of solutions to $ax \equiv b \pmod{m}$ is equal to $\gcd(a,m)$ if and only if $\gcd(a,m)$ divides $b$; else no solutions. Step 3: Compute $d = \gcd(98,49) = 49$. Check if $49$ divides $175$: $$175 / 49 = 3.57 \dots$$ No, actually $49 \times 3 = 147$, remainder $28$ as found. Therefore, $49$ does not divide $175$, hence no solutions exist. Answer: There are 0 solutions, which is not listed. Possibly the user means $\gcd = 7$ by reducing. Alternative approach: Divide entire congruence by $\gcd(98,49) = 49$: $$\frac{98}{49} x \equiv \frac{175}{49} \pmod{\frac{49}{49}}$$ $$2x \equiv 3.57 \pmod{1}$$ Not valid since mod 1 is always 0. Instead, reinterpret: $\gcd(98,49) = 49$ divides 175 as $175 = 49 \times 3 + 28$, so $49$ does **not** divide 175, no solutions. Therefore, no solutions: answer 0. But multiple choice has 5, 6, 7, 8. Double check with another gcd: $\gcd(98,49) = 49$ Since $49 \nmid 175$, no solution. If the problem was $98x \equiv 175 \pmod{98}$, then $\gcd(98,98) = 98$ divides 175? No. Conclusion: No solution. Since no zero answer, user might want number of distinct incongruent solutions when dividing through by gcd. Number of solutions equals $d = \gcd(a,m)$ if $d|b$. Here: $$d = \gcd(98,49) = 49$$ Check divisibility: $$175 \div 49 = 3.5714...$$ not integer So no solutions. Answer: None from given options. Possibly correct intended answer is **0** solutions. 2. Problem 34: Find the last digit of $8^{100}$. Step 1: Last digit depends on $8^n$ mod 10. Step 2: Powers of 8 modulo 10 cycle every 4 steps: $$8^1 \equiv 8 \pmod{10}$$ $$8^2 \equiv 64 \equiv 4 \pmod{10}$$ $$8^3 \equiv 8 \times 4 = 32 \equiv 2 \pmod{10}$$ $$8^4 \equiv 8 \times 2 = 16 \equiv 6 \pmod{10}$$ Cycle: 8, 4, 2, 6 Step 3: Compute $100 \mod 4 = 0$ Step 4: Since remainder is 0, last digit is the last cycle value: 6 Answer: 6 3. Problem 35: Determine which congruence has **no** integral solution. Recall: Congruence $ax \equiv b \pmod{m}$ has solution if and only if $\gcd(a,m)$ divides $b$. Check each option: A. $36x \equiv 432 \pmod{42}$ $$\gcd(36,42) = 6$$ Check if $6 | 432$: $432 \div 6 = 72$ integer Has solution. B. $28x \equiv 13 \pmod{75}$ $$\gcd(28,75) = 1$$ Since gcd is 1, divides all integers, solution exists. C. $32x \equiv 52 \pmod{44}$ $$\gcd(32,44) = 4$$ Check if $4 | 52$: $52 \div 4 = 13$ integer Has solution. D. $21x \equiv 49 \pmod{27}$ $$\gcd(21,27) = 3$$ Check if $3 | 49$: $49 \div 3 \approx 16.33$ not integer No solution. Answer: D Final answers: 33: No solutions (not given; if forced to pick closest, answer is none) 34: 6 35: D