GRE Prep

True/False: is zero NOT an integer

False, it is an integer

True/False: Every integer has a negative counterpart

False, 0 does not have a negative (everything else does)

is infinity an integer?

no

what do you get: a negative times a negative

always positive

what do you get: negative number added to a negative

always negative

what can you get: a negative number subtracted from a negative

can be negative, 0, or positive

what do you get: a negative number divided by a negative

always positive

what do you get: a negative number raised to a positive

can be positive or negative

what do you get: a negative raised to a negative

can get positive or negative

inclusive integer rule (how many integers are there from # to #)

Last - First +1

1 and itself is…

always factors/divisors

nifty finding factor system

start with 1 and #, make gap smaller

how to find the sum of integers in intervals for even

write out first 3 and last 3 to find what the pair sum is, divide by 2, multiply # of pairs by that sum

how to find sum of integers in intervals for odd

write out first 3 and last 3 to find what pair sum is, subtract last number by 1 and divide by 2, whatever that number is subtract by last and add it to first to see what that middle number is, then multiply # of pairs by that sum and then add the middle number

divisibility rules: how to know if 2 is a factor

if the integer ends in 0, 2, 4, or 8

divisibility rule: how to know if 3 is a factor

sum of digits of integer, then look to see if 3 divides evenly into it

divisibility rules: how to know if 4 is a factor

look at last two digits, see if it’s divisible by 4

even ± even =

even

even ± odd =

odd

odd ± odd =

odd

even x even =

even

even x odd =

even

odd x odd =

odd

even / even =

both or neither

even / odd =

even or neither

odd / odd =

odd or neither

even ^ integer =

even except when integer is 0 (even ^0 = 1)

odd ^ integer =

odd

divisibility rule: is it a factor of 5

look at last number, if it is a 0 or 5

divisibility rule: is it a factor of 6

6 is 2×3 so see if it passes both the 2 and the 3 rule

divisibility rule: is it a factor of 8

look at the last 3 digits of number

divisibility rule: is it a factor of 9

same as 3 Rule, add digits of integer and see if 9 can divide into it cleanly

True/False: every integer with a 0 at the end is divisible by 10

True: multiples of 10 end in 0

True/False: Every integer that is divisible by 22 and 33 is divisible by 66

True: as long as that integer can be divided by both 22 and 33, it can be divided by 66.

True/False: Every even integer divisible by 5 is also divisible by 10

The third option would be true since 10=2×5, so you need both a 2 and a 5 to be divisible by 10. Every even integer has a 2

divisibility of 11 rule

do a + then - pattern for each number then combine those values. if it’s a multiple of 11 then it’s divisible

1 is a (multiple/factor) of every integer

factor

0 is a (multiple/factor) or every integer

multiple

sum of 3 consecutive integers

sometimes even, sometimes odd, always multiple of 3

product of 3 consecutive integers

always even, always a multiple of 3, always a multiple of 6, sometimes a multiple of 4 but also 8

consecutive integers

numbers that increase from left to right in order like 1,2,3

when looking at # of factorials and it’s not in prime number, how do you solve?

break it into prime numbers and pick the one that is the limiting factor or that we have less of

what does 0! =

1

what is x^0

1

what is x/0

undefined

which of the following operations CAN result in a negative number:

A.) negative number times negative number

B.) negative number added to negative number

C.) negative number subtracted from negative number

D.) negative number divided by a negative number

E.) negative number raised to a positive

F.) negative number rasied to a negative

B, C, E, F

what are the primes from 1-50 you should remember?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

how to find number of odd factors?

break everything down to their primes, kick out the even numbers, add one to every remaining exponent

TRUE/FALSE: If you consider negative and positive factors, every prime number has four distinct factors

TRUE

TRUE/FALSE: The integer n is guaranteed to have wholly distinct prime divisors from (n+1).

TRUE

TRUE/FALSE: The number of positive factors of integer a^x where a is a prime number, is always equal to x+1

TRUE

TRUE/FALSE

An integer with only one power of 22 will always have an equal number of odd and even positive factors.

how to find GCF with PF?

prime factorization and then what the commonalities are

how to find how many positive factors through GCF

find the GCF (don’t multiply) with PF and then add 1 to all the exponents and multiply those

how to find LCM with PF

use PF, then find the largest exponents between the two of them, write the exponents out and then multiply (go through each term and find max exponent)

what to do for 3 bells ringing type of questions? there are things going at different times, when do they intersect?

find the least common multiple through prime factorization

what is a unit digit?

the last integer of a number, ex: 504, the unit digit would be 4

how to find the unit digit for big exponents like 2³⁶

find the pattern, try 2^1 =2, 2²=4, 2³=8, 2^4= 16, 2^5=32 → this repeats every 4 blocks, 36 is a multiple of 4 meaning its going to be the 4th block so the digit is 6.

what would the remainder of an odd number divided by 2 be

1

how to find the unit digit when you’re adding exponents?

find the unit digit of each number and then add them together

quotient vs. remainder

quotient is what you get from dividing and the remainder is what’s left over

when dividing a negative with a positive, what must the remainder always be

positive or 0

life hack for finding remainders when dividing numbers that don’t evenly split?

divide the numbers, and then multiply the whole number result with your original number, then subtract that by the on you’re diving by → ex: 54/7=7… then do 7×7=49, 54-49=5

how to divide a negative with a positive? ex: -17/5

you have to go left on number line so it would be 5x-4=-20. to get down to -17 you have to add 3 which is your remainder

what to do when y/x but y>x? ex: 5/6?

the answer is always x, so the answer here would be 5

if y/x but y>x, what do you do if you simplify the fraction? what is the remainder? ex: 3/6

you pick the original x so even though 3/6 = 1/2, the remainder is not 1

how to find unit digits if you’re adding exponents?

find the unit digit of each and then add them together!

what happens when you’re looking for the GCF and they have no commonalities after doing prime factorization?

the answer is always 1

what’s the rule when dividing by 10 (could also be in relation to exponents)

the remainder is equal to the unit digit

when dividing high exponents by 5 and the unit digit of the numerator is 0, 1, 2, 3, or 4, what is a good rule of thumb?

answer is unit digit

when dividing high exponents by 5 and the unit digit of the numerator is 6, 7, 8, or 9, what is a good rule of thumb?

find unit digit and then subtract by 5

how to find remainder with high exponent, ex: 3²¹/4?

find the pattern of the first 4 exponents, and then the pattern of what the remainder of each is. 3^1 = 3, ¾ has a remainder of 3, 3²=9, 9/4 has a remainder of 1 the evens are 1 and odds are 3. 21 is an odd number so remainder is 3

exponents and remainders if dividing by 2

if even digit of numerator is even then r=0 but if it’s odd then r=1

how to find remainder of adding fractions that are divided by something

find the remainder separately and add them together. if it’s a larger number than what you’re dividing by do Remainder/Dividing # = final remainder

how to multiply factorials? ex: 7!/8! x 9!/10!

8! is equal to 7! x 8 so then you can cancel the 7!s out and get 1/8. same for other where 10! is equal to 9! x 10, you cancel the 9!s out and get 1/10 × 1/8 = 1/80

how to divide fractions

flip the numerator and denominator of all fractions aside from the first and multiply everything

what if you’re dividing fractions and its stacked like 5/8 / 3/7

flip the bottom and do 5/8 × 7/3 = 35/24

how to divide by 1/something like 6/ 1/5?

same as multiplying, flip the fraction so it becomes 6×5=30

how to tell if something terminates?

do prime factorization, if its simplest forms are a 2 and a 5 then it terminates

terminating vs. non-terminating

terminating means the decimals stop at some point while non-terminating means they go on forever

repeating vs. non-repeating

repeating is when the decimals go on in an endless pattern while non-repeating means they go on with no pattern

how to know if a fraction terminates?

simplify to it’s simplest forms. if there are only 2s, 5s or a combination of both then it terminates

how to turn a repeating decimal into a fraction?

do whatever digit divided by the amount of 9s in the digit. so 37/99

true or false: a fraction can be represented as a non-terminating, non-repeating decimal

false, fractions need to be expressed rationally. if it is non-terminating then it must be repeating

what is rational

can be written as a fraction

TRUE/FALSE: the sum of a rational number and an irrational is irrational

TRUE (must be)T

must be TRUE/FALSE: the sum of two irrational numbers is irrational

not necessarily true

must be TRUE/FALSE: the product of a rational and an irrational number is irrational

FALSE