quant

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

Every positive integer greater than 11 can be written as the product of primes in exactly one way
24=2^3 3^1
100=2^2 5^2
17=17^1
350=2^1 5^2 7^1

ex: 300
1. prime factorize: 2^3 3^1 5^3
2. Add 1 to all exponents: 2^4 3^2 5^4
3. Multiply all exponents: 32
4. 300 has 32 positive factors

numbes you can count (goes up forever)

natural numbers and 0

all whole numbers (and 0) including negative numbers

(last - first) + 1

- Find the number of integers in the interval
- Calculate the sum of the first and last numbers
- Multiply the numbers from steps 1 and 2 together and divide by 2

start w 1, go in order until the gap between the numbers being multiplied is as small as possible

an integer multiplied by all of the positive integers that come before it EX: 4! = 4x3x2x1

Factors of factorials (!) can be larger than the factorial as long as they follow the rules (can be made out of other factors in the factorial).

Prime: Find the prime numbers greater than the factorial
Non-prime: the smallest multiples of the prime numbers greater than the factorial

One prime non-factor of 10! = 11
One non-prime non-factor of 10! = 22

ALL integers are divisable by 1

If it ends in 0,2,4,6,8

If the sum of digits in the integer is divisible by 3

if the sum of the last 2 digits is divisable by 4

An integer is divisible by 5 if the last digit in the number is a 5 or a 0.

An integer is divisible by 6 if it is also divisible by 2 and 3

An integer is divisible by 8 if the last three digits of the number are divisible by 8
*if it ends in 000 it IS divisable by 8

An integer is divisible by 9 if the sum of the digits of the number is divisible by 9.

An integer is divisible by 10 if the last digit in the number is 0.