The theory of numbers, by Robert D. Carmichael ...

8

THEORY OF NUMBERS

These formulas are equivalent in order to the following
five theorems: addition is commutative; multiplication is
commutative; addition is associative; multiplication is associative; multiplication is distributive with respect to addition.
EXERCISES
i. Prove the following relations:
I+2+3+.. + 9-
I+3 —5+. +(21 -I) =n2,
I3+23+33+ * l. 3 = ~ =-(I+2+.. * +n)2.
2
2. Find the sum of each of the following series:
I2+22+3... +l2,
I2+32+53+. * +(2-I)
I'+33+53+... + (2 —I)3.
3. Discover andestablish thelaw suggested by the equations 2 = o+i, 22 = +3,
32=3+6, 42==-6o,..; by the equations I=I3, 3+5=2a, 7+9+II=33,
I3+I3+I7+I9=4,..
~ 2. DEFINITION OF DIVISIBILITY. THE UNIT
DEFIINITIONS. An integer a is said to be divisible by an
integer b if there exists an integer c such that a = bc. It is clear
from this definition that a is also divisible by c. The integers
b and c are said to be divisors or factors of a; and a is said to
be a multiple of b or of c. The process of finding two integers
b and c such that bc is equal to a given integer a is called the
process of resolving a into factors or of factoring a; and a is
said to be resolved into factors or to be factored.
We have the following fundamental theorems:
I. If b is a divisor of a and c is a divisor of b, then c is a
divisor of a.
Since b is a divisor of a there exists an integer p such that
a= b3. Since c is a divisor of b there exists an integer y such
that b=cy. Substituting this value of b in the equation a= bB
we have a=c7y3. But from theorem III of ~ I it follows that
-y/ is an integer; hence, c is a divisor of a, as was to be proved.