if we know a quotient and a remainder, how do we find the divisor and divided numbers?

Question

quotient 22, remainder 12

divisor is greater than remainder, right?

is there a formula??

Answer 1

Yes, there is, however, it will describe an infinite family of answers. If when dividing A by B, we get a quotient Q and remainder R, then by definition, A = BQ + R (B fits into A, Q times with R left over). If you are asked for a specific example of a dividend A and divisor B, notice that B can be any integer larger than R (Why?). Since integers are closed under addition and multiplication, A will come out being an integer. Now try it out on your example Q = 22, R = 12. As a note, the fact that dividing A by B gives a unique quotient Q and remainder R with 0 <= R < B is a theorem called the "Division Algorithm", not to be confused with Long Division.

Answer 2

With the information you have, there are an infinite number of solutions. There is an equation which describes these solutions. There are also boundary conditions.

Let’s call the divisor Y and the dividend X. You have the quotient and remainder given.

X/Y = 22, remainder 12

That remainder 12 term is hard to manipulate. What is a remainder? It is the fraction of Y’s which are left over after the division.

For example, 13/4 = 3, remainder 1 is the same as saying 13/4 = 3 + 1/4

So let’s rewrite our equation as

X/Y = 22 + 12 / Y

Multiplying through by Y on both sides gives:

X = 22Y + 12

This is the equation of a line. However, the whole line isn’t valid. Like you already stated, the divisor must be greater than the remainder:

Y > 12

If the divisor exceeds the remainder, then you just adjust the quotient upwards. For example, if I said that 13/4 = 2, remainder 5, that would be incorrect. The remainder can’t be higher than the divisor.

Another, less obvious boundary is that the dividend is greater than the divisor. X is more than 22 times Y. X > Y

Thus, X > Y > 12 is your boundary.

Draw your Y = 12 line and shade above that. Draw an X = 12 line and shade to the right. and draw a Y = X line and shade the portion where X > Y (the lower portion). The portion of the line X = 22Y + 12 which falls in this boundary contains the solutions to the problem. There might be further boundaries imposed as well that restrict your answers upon this line, since your quotient and remainder are integers. (One of the boundaries is not ever touched. Can you tell which one without graphing?)