Question

Let be any integer and letandbe positive integers. Suppose that when a is divided by, the quotient isqand the remainder isr, so thatand.

If is divided by, show that the quotient isqand the remainderis.

Short answer

It is proved that the quotient isqand remainderis.

Step-by-step solution

The Division Algorithm

Theorem 1.1 states that consider thatas an integer with. Thenandrareunique integersin whichand.

Theorem 1.1 permits for a negative dividend, however, the remaindershould not only be less than the divisor, and it also should be nonnegative.

Show that the quotient is q and remainder r is

Assume that, for.

Multiply the above equation byas follows:

Furthermore, it can be concluded that, because. As a result,can be expressed as a multiple ofwith a remainder from 0 toin this equation.

According to theorem 1.1, the representation is unique; it should be thatis the quotient andis the remainder when dividingby.

Hence, it is proved that the quotient isqand remainderis.