Let's solve each part step-by-step. Recall: \(\text{ord}_p(n)\) is the highest power of \(p\) dividing \(n\), i.e., the exponent of \(p\) in the prime factorization of \(n\). --- ### (a) \(\text{ord}_2(2816)\) **Step 1:** Factor \(2816\) by \(2\): - \(2816 \div 2 = 1408\) - \(1408 \div 2 = 704\) - \(704 \div 2 = 352\) - \(352 \div 2 = 176\) - \(176 \div 2 = 88\) - \(88 \div 2 = 44\) - \(44 \div 2 = 22\) - \(22 \div 2 = 11\) - \(11\) is not divisible by \(2\). **Step 2:** Count the number of divisions: - We divided by \(2\) a total of **8 times**. **Final Answer:** \[ \boxed{\text{ord}_2(2816) = 8} \] --- ### (b) \(\text{ord}_7(2222574487)\) **Step 1:** Divide \(2222574487\) by \(7\) repeatedly: - \(2222574487 \div 7 = 317510641\) - \(317510641 \div 7 = 45358663\) - \(45358663 \div 7 = 6480094.714\) (not integer) **Step 2:** \(2222574487 = 7^2 \times 45358663\), where \(45358663\) is not divisible by \(7\). **Final Answer:** \[ \boxed{\text{ord}_7(2222574487) = 2} \] --- ### (c) \(\text{ord}_p(46375)\) for \(p = 3, 5, 7, 11\) Let's factor \(46375\): #### By \(5\): - \(46375 \div 5 = 9275\) - \(9275 \div 5 = 1855\) - \(1855 \div 5 = 371\) - \(371 \div 5 = 74.2\) (not integer) So, \(5^3\) divides \(46375\). #### By \(3\): - Sum of digits: \(4+6+3+7+5 = 25\), not divisible by \(3\), so \(\text{ord}_3(46375) = 0\). #### By \(7\): - \(46375 \div 7 = 6625\) - \(6625 \div 7 = 946.428...\) (not integer) So, \(7^1\) divides \(46375\). #### By \(11\): - Alternating sum: \(4-6+3-7+5 = -1\). Not divisible by \(11\). So, \(\text{ord}_{11}(46375) = 0\). **Final Answers:** \[ \text{ord}_3(46375) = 0 \\ \text{ord}_5(46375) = 3 \\ \text{ord}_7(46375) = 1 \\ \text{ord}_{11}(46375) = 0 \] --- ## **Summary of Answers** ### (a) \(\boxed{8}\) ### (b) \(\boxed{2}\) ### (c) \[ \boxed{ \begin{aligned} &\text{ord}_3(46375) = 0 \\ &\text{ord}_5(46375) = 3 \\ &\text{ord}_7(46375) = 1 \\ &\text{ord}_{11}(46375) = 0 \end{aligned} } \]