Problem: Find the possible value of p if the remainder when 4p is divided by 9 is 5.

Solution:

To solve this problem, we need to use the concept of remainders. When a number is divided by another number, the remainder is the amount left over after the division is complete. For example, when 10 is divided by 3, the remainder is 1, because 10 = 3 x 3 + 1.

Now, let us consider the given problem:

Step 1: Let p be the positive number we are looking for.

Step 2: According to the problem statement, when 4p is divided by 9, the remainder is 5. This can be written as:

4p = 9q + 5, where q is some integer.

Step 3: Rearranging the equation, we get:

4p - 5 = 9q

Step 4: This implies that 4p - 5 is divisible by 9. Therefore, we can write:

4p - 5 = 9k, where k is some integer.

Step 5: Rearranging the equation, we get:

4p = 9k + 5

Step 6: Now, we need to find the possible values of p. To do this, we can try different values of k and check if the corresponding value of p satisfies the given conditions.

Step 7: Let us start with k = 1. Substituting this in the above equation, we get:

4p = 9 + 5 = 14

Step 8: Solving for p, we get:

p = 14 / 4 = 3.5

Step 9: But p must be a positive integer, and 3.5 is not an integer. Therefore, k = 1 does not give us a valid solution.

Step 10: Let us try the next value of k, which is k = 2. Substituting this in the above equation, we get:

4p = 18

Step 11: Solving for p, we get:

p = 18 / 4 = 4.5

Step 12: Again, p must be a positive integer, and 4.5 is not an integer. Therefore, k = 2 also does not give us a valid solution.

Step 13: Continuing in this manner, we can try different values of k and check if the corresponding value of p is a positive integer. After some trial and error, we find that the possible value of p is:

p = 7

Step 14: We can check that this value satisfies the given conditions. When 4p is divided by 9, the remainder is:

4p = 4 x 7 = 28
28 = 9 x 3 + 1

Step 15: Therefore, the remainder is 1, which is not equal to 5

P = 8