QUANT REVIEW

ARITHMETIC

1.1 Integers

The set of integers, I, is composed of all the counting numbers (i.e., 1, 2,
3, . . .), zero, and the negative of each counting number; that is,
I={. . . ,-3,-2,-1, 0, 1, 2, 3, . . .}.
Therefore, some integers are positive, some are negative, and the integer 0 is
neither positive nor negative. Integers that are multiples of 2 are called even
integers, namely {. . . ,-6,-4,-2,0,2,4,6, . . .}. All other integers are called
odd integers; therefore {. . . ,-5,-3, -1,1,3,5, . . .} represents the set of all
odd integers. Integers in a sequence such as 57, 58, 59, 60, or − 14, − 13, − 12, − 11
are called consecutive integers.

The rules for performing basic arithmetic operations with integers should be
familiar to you. Some rules that are occasionally forgotten include:
(i) Multiplication by 0 always results in 0; e.g., (0)(15) = 0.
(ii) Division by 0 is not defined; e.g., 5 ÷ 0 has no meaning.
(iii) Multiplication (or division) of two integers with different signs yields
a negative result; e.g., (-7)(8)= -56 and (-12)/(4)= -3.
(iv) Multiplication (or division) of two negative integers yields a positive
result; e.g., (-5)(-12)= 60 and (-24)/(-3)=8.
The division of one integer by another yields either a zero remainder, sometimes
called “dividing evenly,” or a positive-integer remainder. For example,
215 divided by 5 yields a zero remainder, but 153 divided by 7 yields a remainder
of 6.



When we say that an integer N is divisible by an integer x, we mean that N
divided by x yields a zero remainder.
The multiplication of two integers yields a third integer. The first two integers
are called factors, and the third integer is called the product. The product is said
to be a multiple of both factors, and it is also divisible by both factors. Therefore,
since (2)(7) = 14,
we can say that
2 and 7 are factors and 14 is the product,
14 is a multiple of both 2 and 7,
and 14 is divisible by both 2 and 7.

Whenever an integer N is divisible by an integer x, we say that x is a divisor
of N. For the set of positive integers, any integer N that has exactly two distinct
positive divisors, 1 and N, is said to be a prime number. The first ten prime
numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
The integer 14 is not a prime number because it has four divisors: 1, 2, 7, and 14.
The integer 1 is not a prime number because it has only one positive divisor.