GRE General Test

## Integers and signs The **integers** are the whole numbers and their negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ Positive integers lie above 0 and negative integers below it, and 0 itself is neither. Adding, subtracting or multiplying integers always gives an integer; dividing one integer by another may not. The sign of a product follows three facts. A positive times a positive is positive, a negative times a negative is positive, and a positive times a negative is negative. So $(-4)(-6) = 24$ and $(-4)(6) = -24$. With more than two factors, the product is negative when an odd number of the factors are negative and positive when an even number are. ## Factors and multiples When integers are multiplied, each of them is a **factor** (or divisor) of the product. Since $84 = 7 \times 12$, both 7 and 12 are factors of 84. The positive factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84, and the negatives of these are factors as well. Factors come in pairs that multiply to 84, which is a reliable way to check a list: $1 \times 84$, $2 \times 42$, $3 \times 28$, $4 \times 21$, $6 \times 14$, $7 \times 12$. A number is a **multiple** of each of its factors, and is **divisible** by them. A few facts follow directly. - Every nonzero integer has infinitely many multiples: 25, 50, 75, and so on. - 1 is a factor of every integer, and 0 is a multiple of every integer. - 0 is a factor of no integer except 0, because nothing multiplied by 0 gives a nonzero number. ###…