The problem provided states:

"When the integer $k$ is divided by 17, the quotient is $p$ and the remainder is 5. When $k$ is divided by 23, the quotient is $q$ and the remainder is 14. Which of the following is true?"

To solve this, we first express the two conditions mathematically:

Step 1: Write the equations

1. When $k$ is divided by 17, the quotient is $p$ and the remainder is 5: $k = 17p + 5$
2. When $k$ is divided by 23, the quotient is $q$ and the remainder is 14: $k = 23q + 14$

Step 2: Set the equations equal

Since both expressions represent $k$, we can set them equal to each other: $17p + 5 = 23q + 14$

Step 3: Simplify the equation

Subtract 5 from both sides: $17p = 23q + 9$ Subtract 23q from both sides: $17p - 23q = 9$

Step 4: Interpret the equation

This equation $17p - 23q = 9$ suggests a relationship between $p$ and $q$. The problem likely provides multiple-choice options asking about which value of $p$ and $q$ satisfy this equation.

However, without the actual options listed in the image, we can only state the equation that needs to be checked against the options.

Summary

The key relationship derived is: $17p - 23q = 9$

To find which of the options is true, you would need to see which set of values for $p$ and $q$ satisfies this equation.

Would you like further details on solving these kinds of problems, or do you have any other questions?

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Here are some related questions to consider:

1. How would the equation change if the remainder when dividing by 17 was different?
2. What happens if both divisors (17 and 23) are changed?
3. Can you find specific integer solutions for $p$ and $q$?
4. How would the problem change if the second remainder was 15 instead of 14?
5. What if the quotient was given as a known integer?
6. How could you use modular arithmetic to solve this problem?
7. What if you only knew one of the two equations?
8. Can this problem be solved graphically or using a different mathematical technique?

Tip: When solving problems involving remainders, expressing the relationships as equations is key. Setting the equations equal when they both describe the same variable allows you to find relationships between the variables involved.