6

––

π

2

446 relatively prime

relatively prime (coprime) Two integers aand b

are relatively prime if their

GREATEST COMMON FAC

-

TOR

is 1. Consequently, the only factor the two num-

bers have in common is 1. For example, 15 and 28 are

relatively prime. Any two consecutive numbers are

relatively prime.

Relatively prime numbers play a key role in the

famous postage-stamp problem:

Which postage values can be obtained using 5-

cent and 7-cent stamps only?

For example, one can compose a postage value of 22

cents with three 5-cent stamps, and one 7-cent stamp,

but the quantity 23 cents cannot be so obtained. One

can check that each quantity 24, 25, 26, 27, and 28 is

possible. Adding multiples of 5 to each of these num-

bers then shows that any quantity greater than 24 is

obtainable. (Below this, one can check that only the

values 5, 7, 10, 12, 14, 15, 17, 19, 20, 21, and 22 can

be composed.)

In general, given two stamp values aand b, with

values aand brelatively prime, one can prove that

there is always a number Nso that every quantity N

and above can be composed as a combination of

stamps of each type. This result is not true if aand b

are not relatively prime.

Mathematicians have shown that the

PROBABILITY

that two integers chosen at random are relatively prime

is ≈0.61.

See also

COMMON FACTOR

;

FACTOR

;

JUG

-

FILLING

PROBLEM

.

remainder theorem The process of

LONG DIVISION

shows that if a

POLYNOMIAL

p(x) of degree nis divided

by a second polynomial q(x) of degree m, one obtains,

apart from a

QUOTIENT

term Q(x),a remainder poly-

nomial R(x) of degree strictly smaller than m:

p(x) = Q(x)q(x) + R(x)

For instance, dividing p(x) = x4– x3+ 2x2+ x+ 4 by

the degree-2 polynomial q(x) = x2+ 1 yields a remain-

der R(x) = 2x+ 1 of degree 1:

In particular, if a polynomial p(x) is divided by a lin-

ear term x– afor some constant a, the result remainder

must be a polynomial of degree zero, that is, a constant:

p(x) = Q(x) (x– a) + b

Setting x= ainto this equation shows that p(a) = 0 + b.

Thus the remainder term bis simply the value of the

polynomial at x= a. This observation is called the

remainder theorem:

If a polynomial p(x) is divided by the term

x– a, for some constant a, then the remain-

der term is p(a):

p(x) = (x– a)Q(x) + p(a)

For example, if p(x) = x3– 7x2+ 2x+ 4 is divided by

x+ 1 = x– (–1), the remainder will be p(–1) = – 1 – 7

– 2 + 4 = –6.

The remainder theorem is useful for finding the

factors of a polynomial. For instance, for the polyno-

mial p(x) = x3– 7x2+ 2x+ 4, we have p(1) = 1 – 7 + 2

+ 4 = 0, indicating that (x– 1) is a factor: x3– 7x2+ 2x

+ 4 = (x– 1)Q(x) + p(1) = (x– 1)Q(x).

See also

FACTOR THEOREM

;

FUNDAMENTAL THEO

-

REM OF ALGEBRA

.

removable discontinuity See

CONTINUOUS FUNCTION

.

Reuleaux’s triangle See

CONSTANT WIDTH

.

Rhind papyrus (Ahmes papyrus) This famous docu-

ment is the oldest written mathematical text known to

exist. Currently housed in the British Museum, this 18-

ft long roll of papyrus, 13 in. wide, dates back to ca.

1650

B

.

C

.

E

. The document was discovered buried in the

Egyptian desert sands, near the Valley of the Kings, in

the mid-1800s by an unknown Egyptian citizen. Visit-

ing Scotsman Alexander Henry Rhind (1833–63), who

had an interest in antiquities, bought the papyrus and

transported it to Britain.

The papyrus is a copy of a work that dates back at

least 200 years earlier. Although the text consists

mainly of numerical problems and solutions, with an

emphasis on practical application, it is clear that the