Quant Group 2

1 divisibility rule

all integers are divisible by 1

2 divisibility rule

an integer is divisible by this integer if it is even (AKA ends in 0, 2, 4, 6, or 8)

3 divisibility rule

an integer is divisible by this integer if the sum of the digits in that integer is also divisible by 3 (ex: 816 → 8 + 1 + 6 = 15, a number divisible by 3)

4 divisibility rule

an integer is divisible by this integer if the last two digits in that integer are also divisible by 4 (example: 56,332 is divisible by 4 because 32 is divisible by 4)

4 divisibility rule

with this divisibility rule, if the last two digits of an integer are 00, the entire number is divisible by 4 because 00 is divisible by 4

5 divisibility rule

an integer is divisible by this integer if the last digit in the number is a 5 or 0

6 divisibility rule

an integer is divisible by this integer if it is also divisible by 2 and 3

8 divisibility rule

an integer is divisible by this integer if the last three digits of the number are divisible by 8

8 divisibility rule

if a number ends in 000, this divisibility rule says it is divisible by 8

9 divisibility rule

an integer is divisible by this integer if the sum of the digits of the number is divisible by 9 (example: 5,985 = 5 + 9 + 8 + 5 = 27 which is a multiple of 9)

10 divisibility rule

an integer is divisible by this integer if the last digit in the number is 0

11 divisibility rule

starting with a + from the left-most digit, assign + and - to adjacent digits of the number, then add the +s and -s separately, and subtract the sum of the -s from the sum of the +s. If the result is a multiple of , the number is divisible by _

prime numbres

numbers with exactly two factors: 1 and the number itself

prime number

2

prime number

3

prime number

5

prime number

7

prime number

11

prime number

13

prime number

17

prime number

19

prime number

23

prime number

29

prime number

31

prime number

37

prime number

41

prime number

43

prime number

47

prime factorization

every positive integer can be written as the product of primes in exactly one way (example: 24 = 2³ x 3^1)

every prime factorization is unique

no two distinct integers share the same prime factorization

determine whether a number is prime

to determine this, take the square root of n and see if it divisible by any prime numbers lower than the square root value

factors of factorials

a factorial (!) has a lot of factors. To save time finding them, convert the factorial into its prime factorization, and then add +1 to every exponent and multiply the exponents together

factors of factorials

example: to find this, cover 8! to its prime factorization, 2^7, 3², 5^1, and 7^1. Then add one to every exponent (so now it is 2^8, 3³, 5², and 7². Then multiply all the exponents together (8 × 3 × 2 × 2)

non-factors of factorials

step 1: find prime numbers greater than factorial number (ex: 20! → 23, 29, 31, 37, 41, etc. are all NOT factors of 20!)

step 2: for non-prime non-factors? find multiples of previous identified primes (ex: 23 → 46, 69, 92) **this trick only works for factorials greater than or equal to 10tra

trailing zeros

the consecutive 0s at the end of an integer; example: 10 has one trailing zero, 100 has two, and 1,000,000 has 6

zero

• 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 of only itself

zero

• if you try to divide by this number, you’ll get an undefined result

• anything raised to the power of this number equals 1

  • n! = 1

primes

these numbers are infinite; can prove this by multiplying any number of prime numbers and adding one

to find the number of positive factors with prime factorization

step 1: prime factorize the integer

step 2: add 1 to each exponent in the prime factorization

step 3: multiply these numbers together

**if a certain integer has only one prime factor, simply add 1 to the exponent to get the total number of positive factors

to find the number of odd factors with prime factorization

step 1: prime factorize the integer

step 2: focus only on odd prime divisors (AKA ignore 2 and its exponent)

step 3: add +1 to every odd prime divisor and multiply exponents together

to find the number of even factors with prime factorization

two ways:

1) # of positive factors - # of odd factors = # of even factors

2) step 1: prime factorize the number

step 2: add +1 to every exponent but DO NOT ADD +1 to 2

step 3: multiply the exponents together

GCF with prime factorization

step 1: prime factorize both numbers

step 2: identify the prime divisors the two numbers share in common & exponents they share in common

step 3: multiply the numbers together (NOT exponents) that they share in common

LCM with prime factorization

step 1: prime factorize both numbers

step 2: write down all of the prime divisors represented across both numbers (so if one is 2²5² and the other is 2²5²11², you would grab all three numbers 2 5 11)

step 3: find largest exponent present for each prime divisor and multiply these together

exponent unit digit patterns

0^n = always 0

1^n = always 1

2^n = 2, 4, 8, 6, 2, 4, 8, 6 and so on → repeats in blocks of 4s

3^n = 3, 9, 7, 1, 3, 9, 7, 1 and so on → repeats in blocks of 4s

4 = 4, 6, 4, 6, 4, 6 and so on → repeats in blocks of 2s

5 = always 5

6 = always 6

7 = 7, 9, 3, 1, 7, 9, 3, 1 → repeats in blocks of 4s

8 = 8, 4, 2, 6, 8, 4, 2, 6 → repeats in blocks of 4s

9 = 9, 1, 9, 1, 9, 1 → repeats in blocks of 2s