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NUMBER THEORY
This broad category is a popular source for SAT questions. At first, students often struggle with these problems since they have forgotten many of the basic properties of arithmetic. So before we begin solving these problems, let's review some of these basic properties.

"The remainder is r when p is divided by q" means p = qz + r; the integer z is called the quotient. For instance, "The remainder is 1 when 7 is divided by 3" means 7 = 3(2) + 1.

Example: When the integer n is divided by 2, the quotient is u and the remainder is 1. When the integer n is divided by 5, the quotient is v and the remainder is 3. Which one of the following must be true?
(A) 2u + 5v = 4
(B) 2u - 5v = 2
(C) 4u + 5v = 2
(D) 4u - 5v = 2
(E) 3u - 5v = 2

Translating "When the integer n is divided by 2, the quotient is u and the remainder is 1" into an equation gives n = 2 u + 1. Translating "When the integer n is divided by 5, the quotient is v and the remainder is 3" into an equation gives n = 5v + 3. Since both expressions equal n, we can set them equal to each other: 2u + 1 = 5v + 3. Rearranging and then combining like terms yields 2u - 5v = 2. The answer is (B).

A number n is even if the remainder is zero when n is divided by 2: n = 2z + 0, or n = 2z.

A number n is odd if the remainder is one when n is divided by 2: n = 2z + 1.

The following properties for odd and even numbers are very useful--you should memorize them:

even x even = even
odd x odd = odd
even x odd = even

even + even = even
odd + odd = even
even + odd = odd

Consecutive integers are written as x, x + 1, x + 2, . . .
Consecutive even or odd integers are written as , x + 2, x + 4, . . .
The integer zero is neither positive nor negative, but it is even: 0 = 2(0).
A prime number is an integer that is divisible only by itself and 1.
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . .
A number is divisible by 3 if the sum of its digits is divisible by 3.
For example, 135 is divisible by 3 because the sum of its digits (1 + 3 + 5 = 9) is divisible by 3.
The absolute value of a number, | |, is always positive. In other words, the absolute value symbol eliminates negative signs.

For example, | -7 | = 7. Caution, the absolute value symbol acts only on what is inside the symbol, | |. For example, -| -7 | = -(+7) = -7. Here, only the negative sign inside the absolute value symbol is eliminated.

Example: If a, b, and c are consecutive integers and a < b < c, which of the following must be true?
I. b - c = 1
II. abc/3 is an integer.
III. a + b + c is even.

(A) I only (B) II only (C) III only (D) I and II only (E) II and III only

Let x, x + 1, x + 2 stand for the consecutive integers a, b, and c, in that order. Plugging this into Statement I yields b - c = (x + 1) - (x + 2) = -1. Hence, Statement I is false.

As to Statement II, since a, b, and c are three consecutive integers, one of them must be divisible by 3. Hence, abc/3 is an integer, and Statement II is true.