GRE arithmetic quant

Exponent rule 2: If you have a negative exponent, you can make it positive by

"flipping" the fraction

To find the number of integers in an interval:

(Last number- First number) +1 For example, to find the number of integers from 27 to 84, inclusive: 84- 27 +1= 58 If you see the word exclusive, you would instead do: 83- 28 +1= 56

Factorials 0-6

0!= 1 1!= 1 2!= 2 3!= 6 4!= 24 5!= 120 6!= 720

Exponent unit digit patterns

0^n= always 0 0^1= always 1 2^n= 2, 4, 8, 6... 3^n= 3, 9, 7, 1... 4^n= 4, 6... 5^n= always 5 6^n= always 6 7^n= 7, 9, 3, 1... 8^n= 8, 4, 2, 6... 9^n= 9, 1...

Exponent rule 5: If you raise anything to the 0 power, it equals

1 except 0^0 which is undefined

Number of multiples in an interval from 1 to n and from n to m where n ≠ 1

1 to n: Divide n by multiple in question and take the whole number result Ex) How many multiples of 31 are there from 1 to 1,000?: 1,000/2 = 32.26 --> answer is 32 n to m where n ≠ 1: ((last multiple- first multiple)/multiple in question) + 1

Sum of integers from n to m, where n ≠ 1

1. Find the number of intergers in the interval 2. Calculate the sum of the first and last number 3. Multiply the numbers from steps 1 and 2 together and divide by 2 Ex) What is the sum of all integers from 30 to 100? 1. 101- 30 +1= 72 2. 30 + 101= 131 3. (72 x 131)/2= 4,716

To find the Greatest Common Factor with Prime Factorization:

1. Prime factorize both numbers 2. Identify the prime divisors the two numbers share in common. Make sure to note the exponents they share in common too. 3. Multiply together what they share in common.

To find the number of positive factors for a given integer using Prime Factorization

1. Prime factorize the integer 2. Add 1 to each exponent in the PF 3. Multiply those numbers together

To find the number of Odd Positive Factors with Prime Factorization for an integer greater than 1:

1. Prime factorize the integer 2. Focus only on the odd prime divisors 3. Apply PF trick to find the number of positive odd factors

To find the number of Shared Factors (positive) with Prime Factorization:

1. prime factorize both numbers 2. Identify what the two numbers share in common 3. Do the "trick" where you add one to each exponent and multiply the numbers together.

To find the Least Common Multiple with Prime Factorization

1. prime factorize both numbers 2. across both numbers, write down all of the prime divisors represented 3. find the largest exponent present for each prime divisor. Write down these exponents and multiply the number out.

The primes up to 50

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47 *2 is the only even prime number

Divisibility rules

3: An integer is divisible by 3 if the sum of the digits in that integer is also divisible by 3. 4: An integer is divisible by 4 if the last two digits in that integer are also divisible by 4. ex) 56, 234, 332 is divisible by 4 because 32 is divisible by 4. Note that if the last two digits are 00, the entire number is divisible by 4 because 00 is divisible by 4. 6: An integer is divisible by 6 if it is also divisible by 2 and 3. 8: An integer is divisible by 8 if the last three digits of the number are divisible by 8. Note that if the number ends in 000, it is divisible by 8. 9: An integer is divisible by 9 if the sum of the digits of the number is divisible by 9.

Three consecutive integers special properties

Adding: The result is ALWAYS a multiple of 3 Multiplying: The product is ALWAYS a multiple of 2, 3, and 6. If the middle consecutive integer is odd, the product is always a multiple of 2, 3, 4, 6, and 8 *apply even if you have Three Consecutive EVEN Integers, or Three Consecutive ODD integers, or Three Consecutive MULTIPLES of 3, ect.

If n^2= 64, n= ____. If we take √64, n= ____.

BOTH -8 and 8 ONLY 8

"Even" roots

Can ONLY take positive values or zero. No negatives allowed. ex) the square root, the 4th root, the 6th root, ect. Can't do √-25

"Odd" roots

Can take positive OR negative values (or zero). Everything is allowed. ex) the cube root, the 5th root, the 7th root, ect. can take positive OR negative values (or zero). Everything is allowed: ∛27 and ∛-27 both make sense.

To find the Total Number of Factors using Prime Factorization:

Do the Positive Factors with PF "trick" and multiply the result by 2

Non-factors of factorials (prime and non-prime)

Ex) Find the smallest integers that are not factors of 20!: 1. Find the prime numbers greater than 20: 23, 29, 31, 37, 41, ect. are all NOT factors of 20 2. If we want to find the non-prime non-factors, find the multiples (greater than the number itself) of our previously identified primes: 23: 46, 69, 92... 29: 58, 87, 116... NOTE that this trick really only works for factorials greater than or equal to 10!

Factors of factorials

Factors of 8!: 1, 2, 3, 4, 5, 6, 7, 8 but ALSO 56 (8 x 7), 30 (6 x 5), 1, 440 (8 x 6 x 5 x 3 x 2)

To find the number of Even Positive Factors with Prime Factorization

Find the total number of positive factors and subtract the number of odd positive factors: number of positive factors - number of odd factors= number of even factors (second way to calculate in notes)

Factors

If an integer is divisible by a certain number, then that certain number is a factor of that integer. For example, the integer 36 is divisible by 4, 1, 9, -6, ect. All of those numbers are factors of 36.

Multiples

If n is an integer, the numbers 0n, 1n, 2n, 3n and so on are multiples of n.

To Convert a Repeating Decimal to a Fraction:

If only one digit repeats, put it over 9. If two digits repeats, put it over 99. If three digits repeats, put it over 999. and so on...

Decimal Number System facts

It is a positional number system, meaning that the location of a digit in a number influences its value. It uses only 10 characters to express an infinite number of numbers: (0-9) It is a "base-10" number system. It uses the digit 0 to indicate "there is nothing in this place."

Properties of 0

It is neither positive nor negative. It is even (because it's divisible by 2). It is a multiple of EVERY integer. It's a factor only of itself. If you try to divide by 0, you'll get an undefined result. Anything raised to the power of 0 equals 1 (exception: 0^0 = undefined). 0!=1

Rational numbers

Numbers that can be written as a fraction.

Irrational numbers

Numbers that cannot be written as a fraction. Any number that, when written as a decimal, is non-terminating and non-repeating.

Infinitude of primes

Primes tend to decrease as numbers get larger

Whole numbers

Similar to natural numbers but 0 is added: 0, 1, 2, 3, 4, 5...

How to determine if the fraction terminates:

Simplify the fraction to its simplest form so that there are no shared factors among the numerator and denominator. If there are only powers of 2 in the denominator, it definitely terminates. If there are only powers of 5 in the denominator, it definitely terminates. If there is a combination of only powers of 2 and powers of 5 in the denominator, it definitely terminates.

Natural numbers

The "counting numbers" For example: 1, 2, 3, 4, 5...

GCF

The Greatest Common Factor (GCF) of two numbers is the largest factor common to both numbers.

LCM

The Least Common Multiple (LCM) of two numbers is the smallest positive multiple they share in common. We have to specifically identify "positive" multiple here because of course 0 is a multiple of every integer.

Integer

The whole numbers and the negative counterparts: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

How to determine whether a number is prime

To determine whether the integer n is prime, take the square root of n and see if it is divisible by any prime numbers lower than that square root value. Ex) To determine if 107 is prime: square root of 107 = 10.34 See if 107 is divisible by primes BELOW that value: 2, 3, 5, and 7. If it's not divisble by any of those smaller primes, then we could conclude that 107 is prime. (It is prime)

Exponent rule 3: If you multiply two numbers together with the same base, you can

add the exponents

A negative subtracted from a negative ____

can result in either a negative number, zero, or a positive number

Sum of multiples in an interval

ex) Find the sum of all multiples of 3 from 1 to 500. : 1. Find the number of multiples of 3 from 1 to 500. : 500/3= 166.67 --> 166 2. List out first three multiples and last three multiples in the interval: 3, 6, 9... 492, 495, 498 3. Notice that every "pair" in that list adds up to 501.: 3 + 498= 501, 6 + 495= 501, 9 + 492= 501 4. We know that every "pair" in the last adds up to 501. We also know that we have a total of 166 multiples of 3, giving us 83 pairs (166/2). We can then find the total sum by multiplying these numbers together.: (166/2) x 501= 41,583 *Still works if you find an odd number of multiples in an interval

Repeating Decimals are always from ____. Non-Repeating Decimals are always from ____

fractions irrational numbers and never fractions

A negative number added to another negative ____

is always negative

Sum of integers from 1 to n

n(n+1)/2

Irrational numbers are ____ (terminating/non-terminating)

non-terminating

Exponent rule 1

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Exponent rule 6: If you multiply two different numbers together with the same exponent, you can

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Exponent rule 7: If you divide two different numbers with the same exponent, you can

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Exponent rule 4: If you divide a number by another number with the same base, you can

subtract the exponents.

Exponent rule 8: If you raise a number with an exponent to another exponent (with parentheses included)

you simply multiply the exponents pic

Remainders and exponents rules

÷1: R is always 0 ÷2: R is either 0 (if number being divided is even) or 1 (if number being divided is odd) ÷3: Write down first one to five remainders and see if you can find some kind of pattern ÷4: Calculate final two digits of number being divided and use divisibility rule with 4 to determine R ÷5: If the unit digit is 0, 1, 2, 3, or 4 --> that's R If the unit digit is 5, 6, 7, 8, or 9 --> R is equal to the unit digit minus 5 ÷6: Try to find some kind of pattern ÷7: Try to find some kind of pattern ÷8: Calculate the final three digits of number being divided and use divisibility rule with 8 to determine R ÷9: Try to find some kind of pattern ÷10: Calculate the unit digit of the number being divided --> that's R