Division Algorithm: Find q & r for a=-5286 and b=19

[Caldus]

If I have to find the quotient q and the remainder r and:

a = -5286
b = 19

How do I go about writing down the steps for this algorithm? I know what the answer will be, but I need to be able to use the division algorithm to prove my answer. Like I know if:

b > 0 and a < 0 (which in this case is true),

then since -a > 0, a = (-q)b - r (where 0 <= r < b).

But how would I know that q is equal to -279 and that the remainder is 15? (Pretending that I didn't the know answer already.)

Thanks.

 

[HallsofIvy]

I have absolutely no idea how you would "know that q is equal to
-279 and that the remainder is 15" since -(-279)(19)-15 is NOT
-5286 (nor is (-279)(19)-15). You have your signs mixed up.

Did you try actually dividing? That is, after all what the division algorithm is!

 

[tacman]

To find the quotient q and remainder r using the division algorithm, follow these steps:

Step 1: Start by writing down the given values for a and b.

a = -5286
b = 19

Step 2: Determine if b is greater than 0 and if a is less than 0. In this case, b > 0 and a < 0, so we can proceed with the algorithm.

Step 3: Since -a > 0, we can rewrite the equation as -a = (-q)b - r.

Step 4: Divide both sides of the equation by b to isolate q. This gives us: q = (-a)/b + (r/b).

Step 5: Since we know that the remainder r must be less than b, we can set an initial value for r as 0 and increase it until we find a value that satisfies the remainder condition (0 <= r < b).

Step 6: Use the given values for a and b to substitute into the equation for q. This gives us: q = (-(-5286))/19 + (r/19).

Step 7: Simplify the equation to get q = 278 + (r/19).

Step 8: Now, we can start increasing the value of r until we find a value that satisfies the remainder condition. In this case, we can see that r = 15 satisfies the condition as it is greater than or equal to 0 and less than 19.

Step 9: Substitute the value of r = 15 into the equation for q to get: q = 278 + (15/19).

Step 10: Simplify to get the final answer: q = 278 and r = 15.

Therefore, the quotient q is -279 and the remainder r is 15. This can be proven by substituting these values back into the original equation: -a = (-q)b - r. We get: -(-5286) = (-(-279))(19) - 15, which simplifies to 5286 = 5286, proving that our values for q and r are correct.