GMAT

## Integers, parity and signs The **integers** are the whole numbers with their negatives: $\dots, -2, -1, 0, 1, 2, \dots$. Zero is an integer, it is even, and it is neither positive nor negative. An integer is **even** if it is a multiple of 2 and **odd** otherwise. Parity follows fixed rules, and they are quicker than trying numbers. | Operation | Result | |---|---| | even + even, odd + odd | even | | even + odd | odd | | even × anything | even | | odd × odd | odd | | odd$^n$ | odd | | even$^n$ (for $n \ge 1$) | even | For signs, a product or quotient of two numbers is positive when they have the same sign and negative otherwise. A product of several non-zero numbers is negative exactly when it has an odd number of negative factors. A number raised to an even power is never negative, so $(-3)^4 = 81$ while $(-3)^3 = -27$. ## Divisibility An integer $a$ is **divisible** by a non-zero integer $b$ when $a \div b$ is an integer, and then $b$ is a **factor** (divisor) of $a$ and $a$ is a **multiple** of $b$. Every integer divides 0. | Divisor | Test on the decimal form | |---|---| | 2 | last digit even | | 3 | digit sum divisible by 3 | | 4 | last two digits divisible by 4 | | 5 | last digit 0 or 5 | | 6 | divisible by both 2 and 3 | | 8 | last three digits divisible by 8 | | 9 | digit sum divisible by 9 | | 10 | last digit 0 | | 11 | alternating digit sum divisible by 11 | For 11, add and subtract the digits alternately from the right. For 2,728: $8 - 2 + 7 - 2 = 11$, so…