Question 573667
Let t and w be the number of tricycles and wagons, respectively. Assuming a tricycle has 3 wheels and a wagon has 4 wheels, we have


*[tex \LARGE 3t + 4w = 36]


This type of equation is called a "linear Diophantine equation." There are several techniques for solving them, but this one is pretty straightforward. Here, we can solve for t:


*[tex \LARGE t = \frac{36 - 4w}{3} = 12 - \frac{4w}{3}].


Obviously, we want t and w to be nonnegative integers. Therefore 4w/3 must be an integer, which implies that w is a multiple of 3. We can list all of the possible solutions:


w = 0, t = 12
w = 3, t = 8
w = 6, t = 4
w = 9, t = 0


These are all of the possibilities. If there were tricycles *and* wagons, then the solutions w = 0, t = 12 and w = 9, t = 0 can be ignored.