Please type Thank you Find q and r as defined in the Divisio

Please type! Thank you.

Find q and r as defined in the Division Algorithm when a = 10 and n = 5.

Solution

The Division Algorithm states: let a be and integer and n be a positive integer. Then there exist unique integers q and r such that a = bq+r and 0<= r < b.

So the basic idea is dividing the number a by number n and then coming up with a quotient q and remainder r. Now, here we have a = 10 and n = 5, and we are interested in finding corresponding q and r. Since q is the quotient so it has to be close to a/b.

So we have equations a = nq+r, we divide this equation by n then it becomes a/n = q + r/n. So, by putting the value of a and n here in the given equation, we get: 10 = 5q + r,

or, 10/5 = q + r/5,

or, 2 = q + r/5,

where it follows that q <= a/n ,i.e., q <= 2 and 0 <= r < b ,i.e., 0 <= r < 5. So it make sense to take q = 2, because this is the biggest integer that it less than or equal to a/n. Thus, the required value of q becomes 2, then we move on to find r by putting r = a - nq, which gives r = 10 - 5*2 = 10 - 10 = 0, i.e., r = 0.

Thus, we get q = 2 and r = 0 for a =10 and n = 5.