ISEE Upper Level

## Integers and the order of operations The **integers** are the whole numbers and their negatives: $\dots, -2, -1, 0, 1, 2, \dots$. The **absolute value** of a number is its distance from 0, so $|-7| = 7$ and $|7| = 7$. Adding a negative number moves left on the number line, and subtracting a negative number moves right, so $5 - (-3) = 5 + 3 = 8$. A product or quotient of two numbers with the same sign is positive, and with different signs it is negative. A product with an even number of negative factors is positive and with an odd number is negative. Operations are done in a fixed order: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication and division share one level, as do addition and subtraction, which is why left to right matters. $$ \begin{aligned} 18 - 4 \times (2 + 1)^2 \div 6 &= 18 - 4 \times 3^2 \div 6 \\ &= 18 - 4 \times 9 \div 6 \\ &= 18 - 36 \div 6 \\ &= 18 - 6 = 12 \end{aligned} $$ A negative base needs parentheses to be squared: $(-3)^2 = 9$, but $-3^2 = -9$ because the exponent applies only to the 3. ## Factors, multiples and primes A **factor** of $n$ divides $n$ with no remainder, and a **multiple** of $n$ is the product of $n$ and an integer. A **prime number** has exactly two positive factors, 1 and itself, so the primes begin 2, 3, 5, 7, 11, 13, 17, 19, 23. The number 2 is the only even prime, and 1 is not prime. Every integer greater than 1 is either…