Number Theory Flashcards
Number Theory Memorization
First twenty-six prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ,41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101
Rules for Prime Numbers
- There are infinitely many prime numbers
- Only positive numbers can be prime
- All prime numbers except 2 & 5 end in 1, 3, 7 or 9
- All prime numbers aboce 3 are the form of 6n-1 or 6n+1
If A is a factor of B and A is a factor of C then…
A is a factor of B+C
If A is a factor of B and B is a factor of C then…
A is a factor of C
Find the number of factors for an integer
- Make prime factorization of integer
- n=ap*bq*cr, where a, b, and c are prime factors and p, q, and r are their powers
- The number of factors of n = (p+1)(q+1)(r+1)
NOTE: this includes 1 and n itself
Greatest Common Factor (Divisor)
- List the prime factors of each number.
- Multiply those factors both numbers have in common. If there are no common prime factors, the GCF is 1.
Example:
54: 2, 3, 3, 3
36: 2, 2, 3, 3
2*3*3=18
Least Common Multiple
- Find the prime factors of each number
- Take out the factors in the second number that repeat from the first
- multiply all remaining factors
mean & median for an evenly spaced set
mean=median=(First+Last)/2
Sum of numbers in an evenly spaced set
mean of the set x number of items in set
How to count consecutive integers
add one before you are done
i.e. How many integers for 6 to 10
10 - 6 = 4 + 1 = 5
How to count consecutive multiples
(Last - First) / increment + 1
How many even numbers from 12 to 24
(24 - 12) / 2 + 1 = 7
Sum of Consecutive Integers
- average the first and last numbers to find precise middle
- count number of terms (one then done)
- average x number of terms
Products of Consecutive Integers & Divisibility
The product of any k consecutive integers is always divisible by k factorial (k!)
Sums of Consecutive Integers and Divisibility
For any set of consecutive integers with an ODD number of items, the sum of all the integers is ALWAYS a multiple of the number of items
For any set of consecutive integers with an EVEN number of items, the sums of all the items is NEVER a multiple of the number of items
As numbers between 0 and 1 are raised to a higher exponent, they…
deacrease
Exponents with compound bases
Exponent may be distributed when multiplying, but not when adding
Adding/Subtracting Exponents
IF EXPONENTS HAVE THE SAME BASE:
add them when multiplying
subtract them when dividing
Nested Exponents
When raising a power to a power combine exponents by multiplying
When an exponent is negative
take the recipricol of the base and make the exponent positive
An Exponent of 0
Any non-zero base raised to the power of zero is 1
Fractional Exponents
the numerator tells us what power to raise the base to and the denominator tells what root to take
An integer is divisible by 4 if…
The integer is divisible by 2, twice
OR
if the last two digits are divisible by by 4
An integer is divisible by 6 if..
The integer is divisible by both 2 and 3
An integer is divisible by 8 if…
The integer is divisible by 2, three times
OR
if the last three digits are divisible by 8