For the two questions visible in the image:

Question 2: You are asked to verify whether a modified Luhn algorithm, which uses "multiply by two, divide the resulting number by 9, and keep the remainder," is equivalent to the original rule of "multiply by two and add the digits."

To check, you should:

1. Analyze the results of multiplying a number $x$ by 2.
2. Observe the digit sum of $2x$.
3. Compare it with dividing $2x$ by 9 and taking the remainder (this relates to modular arithmetic, specifically $2x \mod 9$).

Steps to verify equivalence:

• Compute $2x \mod 9$.
• Compare with the sum of digits of $2x$, which is the original Luhn rule.

You can test the equivalence by trying values of $x$ from 0 to 9.

Question 3: You are tasked to prove the inequality for positive numbers $a_1, a_2, \dots, a_n$:

$\sum_{k=1}^n a_k \leq \sum_{k=1}^n ka_k \leq n \sum_{k=1}^n a_k.$

Proof Outline:

1. First Inequality ($\sum_{k=1}^n a_k \leq \sum_{k=1}^n k a_k$):

   • Since $k \geq 1$ for all terms in the summation, each term $k a_k \geq a_k$.
   • Summing over all $k$, $\sum_{k=1}^n ka_k \geq \sum_{k=1}^n a_k$.

2. Second Inequality ($\sum_{k=1}^n ka_k \leq n \sum_{k=1}^n a_k$):

   • $k \leq n$ for all $k$.
   • So $ka_k \leq n a_k$ for each term.
   • Summing over all $k$, $\sum_{k=1}^n ka_k \leq n \sum_{k=1}^n a_k$.

Would you like me to delve deeper into these proofs or equivalence verification?

Related Questions:

1. What is the significance of modular arithmetic in algorithms like Luhn's?
2. How does the property of digit sums relate to modular arithmetic?
3. Can inequalities like the one in Question 3 be generalized to other scenarios?
4. Are there real-world applications of the inequality in Question 3?
5. What are practical uses of the Luhn algorithm in modern contexts?

Tip: Modular arithmetic often simplifies problems involving remainders, so explore its basics thoroughly!