Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

268 PROPERTIES OF THE INTEGERS [CHAP. 11

The numberqin the above theorem is called thequotient, andris called theremainder. We stress the fact
thatrmust be non-negative. The theorem also states that

r=a−bq

This equation will be used subsequently
Ifaandbare positive, thenqis non negative. Ifbis positive, then Fig. 11-2 gives a geometrical interpretation
of this theorem. That is, the positive and negative multiples ofbwill be evenly distributed throughout the number
lineR, andawill fall between some multiplesqband (q+ 1 )b. The distance betweenqbandais then the
remainderr.

Fig. 11-2

Division Algorithm using a Calculator

Supposeaandbare both positive. Then one can find the quotientqand remainderrusing a calculator as
follows:

Step 1. Divideabybusing the calculator, that is, finda/b.

Step 2. Letqbe the integer part ofa/b, that is, letq=INT(a/b).

Step 3. Letrbe the difference betweenaandbq, that is, letr=a−bq.

EXAMPLE 11.2

(a) Leta=4461 andb=16. We can find that the quotientq=278 and the remainderr=13 by long division,

Alternately, using a calculator, we obtainqandras follows:

a/b= 278. 8125 ..., q= 278 ,r= 4461 − 16 ( 278 )= 13

As expected,a=bq+r, namely:

4461 = 16 ( 278 )+ 13

(b) Leta=−262 andb=3. First we divide|a|=262 byb=3. This yields a quotientq′=87 and a

remainderr′=1. Thus

262 = 3 ( 87 )+ 1

We needa=−262, so we multiply by−1 obtaining

− 262 = 3 (− 87 )− 1

However,−1 is negative and hence cannot ber. We correct this by adding and subtracting the value ofb

(which is 3) as follows:

− 262 = 3 (− 87 )− 3 + 3 − 1 = 3 (− 88 )+ 2

Therefore,q=−88 andr=2.