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.

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 prime or a product of primes in exactly one way (apart from order), which is called its prime factorization.

A factor tree gives it by repeatedly splitting a number: $360 = 36 \times 10 = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5$.

Divisibility tests save division. A number is divisible by

• 2 if its last digit is even;
• 3 if its digit sum is divisible by 3;
• 4 if its last two digits form a number divisible by 4;
• 5 if it ends in 0 or 5;
• 6 if it is divisible by both 2 and 3;
• 9 if its digit sum is divisible by 9;
• 10 if it ends in 0.

The greatest common factor (GCF) of two numbers is the largest number dividing both, and the least common multiple (LCM) is the smallest positive number that both divide. Both come from the prime factorizations: the GCF takes each shared prime to its lower power, and the LCM takes every prime to its higher power. For $36 = 2^2 \times 3^2$ and $60 = 2^2 \times 3 \times 5$:

The product of the two numbers equals the GCF times the LCM, and here $36 \times 60 = 2{,}160 = 12 \times 180$, which is a useful check. The number of positive factors of $n = p^a q^b \cdots$ is $(a+1)(b+1)\cdots$, so 360 has $4 \times 3 \times 2 = 24$ factors.

If $a$ is a factor of $n$ and $b$ is a factor of $m$, then $ab$ is a factor of $nm$, and so $nm$ is a multiple of $ab$. Sums behave differently: if a number divides two others, it divides their sum and their difference.

Fractions

A fraction $\frac{a}{b}$ names $a$ of $b$ equal parts, and $b$ cannot be 0. Equivalent fractions come from multiplying or dividing the top and bottom by the same nonzero number, and a fraction is in simplest form when the top and bottom have no common factor other than 1. A fraction greater than 1 can be written as an improper fraction, $\frac{17}{5}$, or as a mixed number, $3\frac{2}{5}$.

Adding and subtracting need a common denominator, and the least common multiple of the denominators keeps the numbers small:

Multiplying multiplies tops and bottoms, and cancelling common factors first is easier: $\frac{9}{14} \times \frac{7}{12} = \frac{3}{2} \times \frac{1}{4} = \frac{3}{8}$. Dividing by a fraction means multiplying by its reciprocal, which is the fraction turned upside down: $\frac{3}{4} \div \frac{9}{10} = \frac{3}{4} \times \frac{10}{9} = \frac{30}{36} = \frac{5}{6}$. Mixed numbers are turned into improper fractions before multiplying or dividing.

A complex fraction has fractions in its top or bottom. Multiply the top and bottom by a common denominator, or divide the top by the bottom:

Decimals and comparing numbers

Place value in a decimal continues to the right of the point as tenths, hundredths, thousandths. To add or subtract, line up the points. To multiply, ignore the points, multiply the whole numbers, then place the point so that the answer has as many decimal places as the two factors together: $0.4 \times 0.03$ has $1 + 2 = 3$ places, and $4 \times 3 = 12$, so the product is $0.012$. To divide by a decimal, move the point the same number of places in both numbers until the divisor is whole.

A fraction becomes a decimal by dividing, so $\frac{3}{8} = 0.375$. A decimal that stops is written over a power of 10 and simplified, so $0.36 = \frac{36}{100} = \frac{9}{25}$. Fractions whose simplest denominators have only the prime factors 2 and 5 give decimals that stop; the others repeat, as in $\frac{1}{3} = 0.\overline{3}$ and $\frac{2}{7} = 0.\overline{285714}$.

To compare fractions, rewrite them over a common denominator, or cross-multiply: $\frac{5}{9}$ and $\frac{4}{7}$ compare as $5 \times 7 = 35$ against $4 \times 9 = 36$, so $\frac{5}{9} < \frac{4}{7}$. Comparing each with $\frac{1}{2}$ often settles it faster. On the number line, negative numbers reverse the usual order of size, so $-\frac{1}{3} > -\frac{1}{2}$.

Estimating and rounding

To round, look at the digit just after the place being kept: 5 or more rounds up, and 4 or less leaves the digit. Estimation rounds every number to one or two easy digits before calculating, and the result tells which answer choice is reasonable: $48.7 \times 0.52 \approx 50 \times 0.5 = 25$. Rounding one number up and another down where they are divided or subtracted keeps errors from compounding. Square roots are estimated between perfect squares, so $\sqrt{50}$ lies between $\sqrt{49} = 7$ and $\sqrt{64} = 8$, and is just above 7.