If x is a positive integer, r is the remainder when x is divided by 4, and R is the remainder when x is divided by 9, what is the greatest possible value of r2 + R ?a)25b)21c)17d)13e)11Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To find the greatest possible value of r² + R, we need to consider the maximum remainders for r and R.

When x is divided by 4, the maximum remainder is 3 (since x is a positive integer). Therefore, the maximum value for r is 3.

When x is divided by 9, the maximum remainder is 8 (since x is a positive integer). Therefore, the maximum value for R is 8.

Substituting these values into the expression r² + R:

r² + R = 3² + 8 = 9 + 8 = 17

Therefore, the greatest possible value of r² + R is 17.

The correct answer is C: 17.

Answer

Understanding the problem:
Given x is a positive integer, r is the remainder when x is divided by 4, and R is the remainder when x is divided by 9. We need to find the greatest possible value of r^2 + R.

Solution:
To find the greatest possible value of r^2 + R, we need to find the maximum values of r and R.

Finding the maximum value of r:
The possible remainders when a number is divided by 4 are 0, 1, 2, and 3. To maximize r^2, we need to choose the largest possible remainder, which is 3. Therefore, r = 3.

Finding the maximum value of R:
Similarly, the possible remainders when a number is divided by 9 are 0, 1, 2, 3, 4, 5, 6, 7, and 8. To maximize R, we need to choose the largest possible remainder, which is 8. Therefore, R = 8.

Calculating the maximum value of r^2 + R:
Substitute r = 3 and R = 8 into the expression:
r^2 + R = 3^2 + 8 = 9 + 8 = 17.

Therefore, the greatest possible value of r^2 + R is 17, which corresponds to option 'C'.

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