Gre math Review 1 Arithmetic

1.1 Integers

The integers are the numbers 1, 2, 3, and so on, together with their negatives,  negative 1, negative 2, negative 3, dot dot dot, and 0.

Thus, the set of integers is   dot dot dot, negative 3, negative 2, negative 1, 0, 1, 2, 3, dot dot dot.

The positive integers are greater than 0, the negative integers are less than 0, and 0 is neither positive nor negative. When integers are added, subtracted, or multiplied, the result is always an integer; division of integers is addressed below. The many elementary number facts for these operations, such as

7 + 8 = 15,
  78 minus 87 = negative 9,
  7 minus negative 18 = 25, and
  7 times 8 = 56,
should be familiar to you; they are not reviewed here. Here are three general facts regarding multiplication of integers.

Fact 1: The product of two positive integers is a positive integer.

Fact 2: The product of two negative integers is a positive integer.
Fact 3: The product of a positive integer and a negative integer is a negative integer.

When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. For example,   2 times 3 times 10 = 60,

so 2, 3, and 10 are factors of 60. The integers 4, 15, 5, and 12 are also factors of 60, since   4 times 15 equals 60 and 5 times 12 = 60.
The positive factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The negatives of these integers are also factors of 60, since, for example,   negative 2 times negative 30 = 60.
There are no other factors of 60. We say that 60 is a multiple of each of its factors and that 60 is divisible by each of its divisors. Here are five more examples of factors and multiples.

Example A: The positive factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Example B: 25 is a multiple of only six integers: 1, 5, 25, and their negatives.
Example C: The list of positive multiples of 25 has no end: 0, 25, 50, 75, 100, 125, 150, etc.; likewise, every nonzero integer has infinitely many multiples.
Example D: 1 is a factor of every integer; 1 is not a multiple of any integer except   1 and negative 1.
Example E: 0 is a multiple of every integer; 0 is not a factor of any integer except 0.

The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of both a and b. For example, the least common multiple of 30 and 75 is 150. This is because the positive multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, etc., and the positive multiples of 75 are 75, 150, 225, 300, 375, 450, etc. Thus, the common positive multiples of 30 and 75 are 150, 300, 450, etc., and the least of these is 150.

The greatest common divisor (or greatest common factor) of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b. For example, the greatest common divisor of 30 and 75 is 15. This is because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, and the positive divisors of 75 are 1, 3, 5, 15, 25, and 75. Thus, the common positive divisors of 30 and 75 are 1, 3, 5, and 15, and the greatest of these is 15.

When an integer a is divided by an integer b, where b is a divisor of a, the result is always a divisor of a. For example, when 60 is divided by 6 (one of its divisors), the result is 10, which is another divisor of 60. If b is not a divisor of a, then the result can be viewed in three different ways. The result can be viewed as a fraction or as a decimal, both of which are discussed later, or the result can be viewed as a quotient with a remainder, where both are integers. Each view is useful, depending on the context. Fractions and decimals are useful when the result must be viewed as a single number, while quotients with remainders are useful for describing the result in terms of integers only.

Regarding quotients with remainders, consider two positive integers a and b for which b is not a divisor of a; for example, the integers 19 and 7. When 19 is divided by 7, the result is greater than 2, since  2 times 7 is less than 19, but less than 3, since   19 is less than 3 times 7. Because 19 is 5 more than   2 times 7, we say that the result of 19 divided by 7 is the quotient 2 with remainder 5, or simply 2 remainder 5. In general, when a positive integer a is divided by a positive integer b, you first find the greatest multiple of b that is less than or equal to a. That multiple of b can be expressed as the product qb, where q is the quotient. Then the remainder is equal to a minus that multiple of b, or   r = a minus qb, where r is the remainder. The remainder is always greater than or equal to 0 and less than b.

Here are three examples that illustrate a few different cases of division resulting in a quotient and remainder.

Example A: 100 divided by 45 is 2 remainder 10, since the greatest multiple of 45 that’s less than or equal to 100 is   2 times 45, or 90, which is 10 less than 100.
Example B: 24 divided by 4 is 6 remainder 0, since the greatest multiple of 4 that’s less than or equal to 24 is 24 itself, which is 0 less than 24. In general, the remainder is 0 if and only if a is divisible by b.
Example C: 6 divided by 24 is 0 remainder 6, since the greatest multiple of 24 that’s less than or equal to 6 is   0 times 24, or 0, which is 6 less than 6.

Here are five more examples.

Example D: 100 divided by 3, is 33 remainder 1, since
  100 = 33 times 3, + 1.
Example E: 100 divided by 25 is 4 remainder 0, since
  100 = 4 times 25, + 0.
Example F: 80 divided by 100 is 0 remainder 80, since
  80 = 0 times 100, + 80.
Example G: When you divide 100 by 2, the remainder is 0.
Example H: When you divide 99 by 2, the remainder is 1.

If an integer is divisible by 2, it is called an even integer; otherwise it is an odd integer. Note that when a positive odd integer is divided by 2, the remainder is always 1. The set of even integers is  dot dot dot, negative 6, negative 4, negative 2, 0, 2, 4, 6, dot dot dot,

and the set of odd integers is  dot dot dot, negative 5, negative 3, negative 1, 1, 3, 5, dot dot dot.

Here are six useful facts regarding the sum and product of even and odd integers.

Fact 1: The sum of two even integers is an even integer.
Fact 2: The sum of two odd integers is an even integer.
Fact 3: The sum of an even integer and an odd integer is an odd integer.
Fact 4: The product of two even integers is an even integer.
Fact 5: The product of two odd integers is an odd integer.
Fact 6: The product of an even integer and an odd integer is an even integer.

A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself. 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, since it has four positive divisors: 1, 2, 7, and 14. The integer 1 is not a prime number, and the integer 2 is the only prime number that is even.

Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors. Such an expression is called a prime factorization. Here are six examples of prime factorizations.

Example A:   12 = 2 times 2 times 3, which is equal to 2 to the power 2, times 3

Example B:   14 = 2 times 7
Example C:   81 = 3 times 3 times 3 times 3, which is equal to 3 to the 4th power
Example D:   338 = 2 times 13 times 13, which is equal to 2, times the quantity 13 to the power 2
Example E:   800 = 2 times 2 times 2 times 2 times 2, times, 5 times 5, which is equal to 2 to the power 5, times 5 to the power 2
Example F:   1,155 = 3 times 5 times 7 times 11

An integer greater than 1 that is not a prime number is called a composite number. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.