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Suppose we have the following program that computes the quotient and remainder when dividing a by b: r = a q 0 while r >= b: r = r b q += 1 and precondition: (a > 0) ^ (6 > 0) postcondition: (a = bq+r) 1 (0

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The given program computes the quotient and remainder when dividing `a` by `b` using a loop.

Let's break down the program step by step:

1. Initialize variables: `r`, `q`, and `a` are initialized to the value of `a`, and `b` is initialized to 0.

2. Enter the loop: The loop condition `r >= b` is checked. If the condition is true, the loop body is executed; otherwise, the loop terminates.

3. Update `r`: `r` is updated by subtracting `b` from it.

4. Update `q`: `q` is incremented by 1.

5. Repeat steps 2-4 until `r` is less than `b`.

6. Terminate the loop: When `r` is less than `b`, the loop terminates, and the program proceeds to the next line.

7. Postcondition: The final values of `a`, `b`, `q`, and `r` satisfy the equation `a = bq + r`, where `q` is the quotient and `r` is the remainder.

Let's consider an example:

Suppose `a = 15` and `b = 3`.

1. Initialize variables: `r = 15`, `q = 0`, `a = 15`, and `b = 3`.

2. Enter the loop: Since `r` (15) is greater than or equal to `b` (3), the loop body is executed.

3. Update `r`: `r` is updated to `r - b` = 12.

4. Update `q`: `q` is incremented by 1, becoming 1.

5. Repeat steps 2-4: `r` is updated to 9, `q` remains 1.

6. Repeat steps 2-4: `r` is updated to 6, `q` remains 1.

7. Repeat steps 2-4: `r` is updated to 3, `q` remains 1.

8. Repeat steps 2-4: `r` is updated to 0, `q` remains 1.

9. Terminate the loop: Since `r` is now less than `b`, the loop terminates.

10. Postcondition: The final values of `a`, `b`, `q`, and `r` satisfy the equation `a = bq + r` as `15 = 3 * 1 + 0`.

So, when `a = 15` and `b = 3`, the quotient `q` is 1, and the remainder `r` is 0.

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