Terms and tricks Flashcards
distinct
A distinct element is a unique digit, number, or integer in a number, set, list, or any other grouping of digits or numbers. For example, 10,899 has four distinct digits and the set {1, 3, 4, 1, 8, 3} has four distinct numbers.
even and odd
A trick for checking work and/or quickly eliminating answers. When adding or subtracting, if both numbers are even or both are odd, you get an even result; if mixed, you get an odd. When multiplying or dividing (without remainder– this doesn’t hold for fractions), if either number is even, you get an even result; only if both are odd do you get an odd.
consecutive integers
For the purposes of the GRE, integers are considered consecutive if they are evenly spaces. So, -1, 0, 1, 2 are consecutive, but so are -8, 1, 10, 19. In short, if you can write out the set using series notation (e.g. the second series could be written as, for i = 0 to 3, n = 9i - 8).
divisibility
Integers are divisible by their factors. Tricks:
3: sum of digits is divisible by 3
4: last 2 digits are divisible by 4
6: even and divisible by 3
8: last 3 digits are divisible by 8
9: sum of digits is divisible by 9
remainder
Always less than the divisor (e.g. if dividing by 6, the remainder must be [0, 5]). If the divisor is larger than the dividend, the remainder is simply the dividend (i.e. the number being divided). For example, the remainder of 5/6 is 5.
factor
A factor of an integer is a number that divides evenly (i.e. without remainder) into that integer. When listing factors, do it in pairs, starting with 1 and the number itself, followed by the next smallest factor (e.g. 2, if even) and it’s pair, etc. For example, 24: 1, 24, 2, 12, 3, 8, 4, 6. Once done, you can reorder them if desired.
multiples
The multiples of an integer are all the integers that are a product of that integer and another integer, which is always an infinite set (except for 0, whose only multiple is 0). For example, the multiples of 8 are: 0, 8, 16, 24, 32…. Zero is a multiple of every number.
prime
An positive integer whose only factors are itself and 1. 2 is the first prime, as well as the only even one. (Note: this means 1 is NOT a prime, something the GRE writers will sometimes implicitly test. Also, negative primes exist in some branches of math, but not for the purposes of the GRE.)
P|E|MD|AS
A mnemonic for the basic four categories for order of operations: parentheses, exponents, multiplication & division, addition & subtraction. Within each category, evaluate left-to-right.
adding or subtracting fractions
If the denominators are the same, just add or subtract the top numbers and keep the denominator. If different, use the bowtie method: multiply each numerator by the other fraction’s denominator and the denominator of each term becomes the product of the two denominators. e.g. 2/3 + 3/4 = 8/12 + 9+12 = 17/12 or 5/7 - 9/2 = 10/14 - 63/14 = -53/14.
multiplying fractions
Just multiply the numerators, then multiple the denominators, and reduce the result. e.g. 4/5 * 11/12 = 44/60 = 11/15. Alternatively, you can reduce before multiplying diagonally (e.g. in the example above, you could have divided the numerator of 4/5 and the denominator of 11/12 by their common factor 4, then multiplied 1/5 * 11/3 = 11/15).
dividing fractions
Flip the second fraction (i.e. change the divisor to its reciprocal), then multiply. e.g. 2/3 ÷ 4/5 = 2/3 * 5/4 = 10/12 = 5/6
comparing fractions
If both fractions are positive and the numerators are the same, larger denominator => smaller number. If both positive and denominators are the same; larger numerator => larger number. If both negative, it’s the opposite– larger denominator => larger number; larger numerator => smaller number. If the numerators and denominators are both different, just convert the fractions to ones with common denominators using the bowtie method and compare them. Converting the fractions to decimals makes everything extra clear, so use the calculator if unsure.
improper fraction
A fraction with a numerator larger than the denominator, e.g. 5/2
word problem symbol translations
percent or % => /100
is => =
of, times, product => *
what (or any unknown) => replace with a variable (x, y, a, etc.)
percent change
The percentage by which something has increased or decreased, equal to difference / original. On % increase problems, the original is the smaller number; on % decrease problems, it’s the larger number. On the GRE, percent change is never stated as a negative number; instead, the problem will ask for a percent decrease, which should be stated as a positive number. For example, the percent decrease from 80% to 60% is 25% (i.e. (80-60)/80).
associative law
when adding or multiplying a series of numbers, you can regroup or reorder the numbers in any way you like without changing the result. e.g. 4 + (5 + 8) = 5 + (4 + 8) = 8 + 4 + 5 and (ab)(cd) = a(bcd) = d(cab).
distributive law
Multiplying a number by a group of numbers added together is the same thing as doing each multiplication separately before adding the results. e.g. a(b + c) = ab + ac and a(b - c) = ab - ac. This can make math easier. For example: 12(66) + 12(24) = 12(66 + 24) = 12(90) = 1080
decimals
Terminating decimals are non-repeating (i.e. they end, like 0.25). Repeating decimals never end, but repeat one or more digits infinitely. For instance, 1/3 is the repeating decimal 0.33333…
zero
For the purposes of the GRE, zero is an integer that is neither positive or negative. It is a multiple of every integer.
absolute value
A number’s distance away from zero on the number line
last digit of x^n
The units digit of any integer x raised to a natural number (i.e. positive integer) n can be found in the following way. x^n for n = 1…infinity will produce a repeating pattern of units digits that is either 1, 2, or 4 digits in length, depending on what the units digit of x is. Here are the patterns for the 10 possible units digits of x:
0, 0, 0, 0…
1, 1, 1, 1…
2, 4, 8, 6…
3, 9, 7, 1…
4, 6, 4, 6…
5, 5, 5, 5…
6, 6, 6, 6…
7, 9, 3, 1…
8, 4, 2, 6…
9, 1, 9, 1…
To reproduce this list, just start with the units digit, u and square it. If you get the same units digit, you’re done– it will repeat forever. If you get a different units digit, multiply it by u until you get the original units digit again or you have a sequence of 4 (including u as the first in the sequence). The fifth in the sequence will always match u. Stated mathematically, for all integers, x: x % 10 == x^5 % 10. Here’s an example: x = 2394583. 33 = 9 (not the same, so keep going). 93 = 27 (not the same, so keep going), 7*3 = 21, so the units digits of x^n will be 3, 9, 7, 1, which will repeat forever if we keep going. If we want the units digit of 2394583^39431732, we just take the exponent %4, which is 0, so we take the 4th digit in our repeating pattern, which is 1.
inequalities
When dealing with questions were inequalities are listed in the problem, be EXTRA careful to keep track of whether they are <= and >=, or < and >. This is especially important when dealing with integers. Often in the same problem statement, the GRE will mix these symbols together, such as -8 <= a < 3 and 4 < b <= 29. In this case, a is [-8, 2] and b is [5, 29]. Be extra cognizant of the boundary cases and exactly what sign is used when inequalities are involved. Also be careful with English representations of edge cases. “Less than”, “more than”, and “between” are not inclusive (i.e. they represent <, >, and < >, respectively) unless the word “inclusive” is used.
inequalities including absolute values
|x| < y can be rewritten as two equations without the absolute value function:
x < y, for x >= 0
x > -y, for x < 0
Not that for negative values of x, you reverse both the direction of the inequality AND the sign of the opposite side of the equation. So if the problem is: Find all integers, x, where |x+8| < 3, this would expand to :
(x+8) < 3 for (x+8) >= 0 => x < -5 for x >= -8, which gives us -8, -7, -6
(x+8) > -3 for (x+8) < 0 => x > -11 for x < -8, which gives us -9 and -10.
Taken together, we have the set {-6, -7, -8, -9, -10} as our answer.
