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
An integer is divisible by 9 if…
the sum of the integers digits is divisible by 9
Find factor pairs for an integer
- make a table with 2 columns labeled “small” and “larger”
- start with 1 in the small column and integer in th large column
- Test the next possible factor and repeat until the numbers in the small and large columns run into each other
First and only even prime
2
Even +/- Even
Even
Odd +/- Odd
Even
Odd +/- Even
Odd
Odd x Odd
Odd
Even x Even
Even
Odd x Even
Even
Even/Even
Even, odd or non integer
Even/Odd
Even or non integer
Odd/Even
non integer
odd/odd
odd or non integer
sum of two prime numbers greater than two will be
even,
so, if sum of two unknown prime numbers is odd one of those primes MUST be 2
Consecutive Integers (define)
increments of 1
these are also considered consecutive multiples and evenly spaced sets
Consecutive Multiples (define)
Multiples of the increment (multiples of 3: 3,6,9,12)
These are also considered evenly spaced sets
Evenly Spaced Sets (define)
Constant increments:
example: 2,5,8,11,14
evenly spaced sets (properties)
mean and median are equal
mean and median are average of firstand last terms
sum of elements in the set equals:
mean x number of items in set
number of items in set is:
last - first + 1, for consecutive integers
last - first/ increment +1, for consecutive multiples
Factor Foundation Rule as it applies to the product of k consecutive integers
product of k consecutive integers is always divisible by k!
any product of 3 consecutive integers will have at least one multiple of 3 and one multiple of 2
any product of 4 consecutive integers will have at least one multiple of 4 and one multiple of 3 and one multiple of 2
etc. etc.
Sums of CONSECUTIVE INTEGERS and divisibility
for any set of consecutive integers with an odd number of items, the sum of al the integers is always a multiple of the number of items
for any set of consecutive integers with an even number of items, the sum of all the items is NEVER a multiple of the number of items
SUM= average x number of items
For odd numbers of items the average is an integer and for even numbers it is not
x2
x or -x
with a positive exponent we cannot be sure which it is.
we must be told x is positive to affirm that it is.
exponent with a base of 0
0
exponent with a base of 1
1
x6 = x7 = x15
x must equal 0 or 1
(only because of the positive exponent, if they were all negative x could also equal -1)
exponents of a positive proper fraction
as the exponent increases the value of the expression decreases
product as compound base with exponent
either multiply the numbers in the base or distribute the exponent
sum as a compound base with an exponent
you MUST add the numbers together before applying the exponent
(you cannot distribute the exponent)
Exponents: multiplying terms with the same base
Add the exponents
34 x 32 = 3(4+2) =36
Exponents: dividing terms with the same base
subtract the exponents
36 / 32 = 3(6-2) =34
When raising a power to a power (nested exponents)
Combine exponents by multiplying
negative exponent
take the reciprocal of the base and change the sign of the exponent to a positive
any number without an exponent…
has an implied exponent of 1
any non zero based raised to 0
1
fractional exponents
numberator tells us what power t oraise the base to and the denominator tells us which root to take
xa * xb
xa+b
c3 * c5
c8
35 * 38
313
5(5n)
51(5n)
5n+1
ax * bx
(ab)x
24 * 34
64
125
210 * 35
xa/xb
xa-b
25/211
1/26
2-6
x10/x3
x7
(a/b)x
ax/bx
(10/2)6
106/26
56
35/95
(3/9)5
(1/3)5
(ax)y
axy
(ay)x
(32)4
32*4
38
34*2
(34)2
x-a
1/xa
(3/2)-2
(2/3)2
4/9
2x-4
2/x4
xa/b
b√ xa
(b√x)a
274/3
³√274
(³√27)4
34
81
5√x15
X3
ax + ax + ax
3ax
34 + 34 + 34
3 * 34
35
3x + 3x + 3x
3 * 3x
3x+1
Simplifying exponential expressions
you can simplify exponentialexpressions that are linked by multiplication or division
you CANNOT simplify expresions linked by addition or subtraction (though they can sometimes be factored)
example
74 + 76 = 74(1 + 72)
odd roots
will have the same sign as the base
even roots
will only have a positive value
√2
1.4
√3
1.7
√5
2.25
√121
11
112
121
122
144
√144
12
132
169
√169
13
142
196
√196
14
152
225
√225
15
√256
16
162
256
202
400
√400
20
252
625
√625
25
302
900
√900
30
23
8
³√8
2
33
27
³√27
3
43
64
³√64
4
53
125
³√125
5
n√x / n√y
n√x/y
n√x * n√y
n√xy
b√xa
(b√x)a
xa/b
all perfect squares have an ____ number of total factors
odd
the prime factorization of a perfect square contains only ____ number of primes
even
length of a number
the number of primes (not necessarily distinct primes)
if a prime factorization contains an odd number of primes
it is not a perfect square
if a number is a perfect cube ten itis fomed from…
primes in sets of three
if k3 is divisible by 240, what is the least possible value of integer k
60
N! is a multiple of
all integers for 1 to N
An = 3n +7
a set of numbers 1 - nth will have what formulas
sum = (average) * number in set
= [(solve for 1) + sequence formula] / 2 \* n
= [(3\*1 +7) + 3n +7] /2 \* n
= (10 + 3n + 7)/2 \* n
= solve
range of a set
first number - last number
sum of n consecutive integers
when n is odd
is divisible by n
sum of n consecutive integers
when n is even
is NOT divisible by N
the sign of an integer is unclear when….
keep this in mind on data sufficiency problems!!!!
it is raised to a positive power
absolute value of integer
is ab< 0 ?
|a| * b <0
(this means b is negative but since we do not know if a is negative, we cannot answer the question)
a4 * b < 0
(this means b is negative but the positve power hides te sign of a, if a is negative, then ab is positive, is ab>0 positive, then ab <0)
if a/b yields a remainder of 5 & c/d yields a remainder of 8, and a, b, c, and d are integers, what is the smallest possible value of b+d
remainders must be smaller than the divisor
5<b 8<d
6+9 = 15
if x leaves a remainder of 4 after division by 7 and y leaves a remainder of 2 after divsion by 7 what is the remainder of x+y/7
6
you can add and subtract remainders, directly, as long as you correct excess or negative remainders
if x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7 then what is the remainder of x+z/7
you can add and subtract remainders as long as you correct excess or negative remainders
4+5=9 - 7 = 2
if x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7 then what is the remainder of x-z/7
you can add and subtract remainders as long as you correct excess or negative remainders
4-5 = -1 + 7 = 6
if x has a remainder of 4 when divided by 7 and z has a remainder of 5 when divided by 7 what is the remainder when x*z is divided by 7
you can multiply remainders as long as you correct excess remainders at the end
4*5 = 20 -(7*2)= 6
remainders and decimals
how to convert the decimal portion into a remainder
x (remainder) /divisor=decimal
example: 17/5=3.4
.4 = x/5
when add or subtracting two numbers, neither of which is divisible by 2… the result will
always be divisible by 2
write an arbitrary odd integer algebraically
2n + 1
absolute value | x- y |, define
distance between x and y
x3 - x is the same as
and if
x3 - x = p and x is odd, and x >1, is p divsible by 24
x(x2 - 1)
x(x - 1)(x +1)
consecutive integers
since x is odd, x-1 and x+1 are even
each divisible by 2 one at least one divisible by 4,
4x2=8
in a set of 3 consecutive integers at least one will be divisible by 3
3x8=24, YES
x2 - x
x(x-1)
x4 - x2
x2(x2-1)
x2(x+1)(x-1)
75 - 73
73(72 - 1)
48 * 73
58 + 59 +510
58(1+5+52)
31 * 58
z3 - z
z(z2-1)
z(z+1)(z-1)
10(b+1)
10(10b)
10(b-1)
10b/10
35 + 35 + 35
3(35)
36
ab - ab-1
ab(1 - a-1)
ab-1(a - 1)
pq + pr + qs +rs
p(q + r) + s(q+r)
(p+s)(q+r)
adding or subtracting roots
roots act like variables, you can only combine them if they are like terms
simplify roots
√80 - √45
4√5 - 3√5 = √5
how to rationalize a denominator with square roots
use conjugates to cancel out the square root
the sum of two multiples of a number
is a multiple of that number
example :
a multiple of 3 + another multiple of 3 = multiple of 3
12 + 9 = 21
How to determine if a r/s will result in a terminating decimal
- you must know s
- the remainder of r/s must be less than s
- put all possible remainders of s over s and find out if any of those are terminating decimals
example s=4
only possible remainders are 1, 2 and 3
1/4 = .25, 2/4 = .5, 3/4 = .75 – so yes, r/s in this case will have a terminating decimal
the sum of all distinct factors of a perfect square is always …
odd
example
4: 2 + 4 + 1 = 7
9: 3 + 9 + 1 = 13
convert to decimal and percent
1/2
.5
50%
convert to decimal and percent
1/3
.33
33%
convert to decimal and percent
2/3
.66
66%
convert to decimal and percent
4/6
.66
66.6%
convert to decimal and percent
2/6
.33
33%
convert to decimal and percent
3/6
.5
50%
convert to decimal and percent
4/6
.66
66.6%
convert to decimal and percent
5/6
.83
83%
convert to decimal and percent
1/4
.25
25%
convert to decimal and percent
2/4
.5
50%
convert to decimal and percent
3/4
.75
75%
convert to decimal and percent
1/5
.2
20%
convert to decimal and percent
2/5
.4
40%
convert to decimal and percent
3/5
.6
60%
convert to decimal and percent
4/5
.8
80%
convert to decimal and percent
1/8
.125
12.5%
convert to decimal and percent
2/8
.25
25%
convert to decimal and percent
3/8
.375
37.5%
convert to decimal and percent
4/8
.5
50%
convert to decimal and percent
5/8
.625
62.5%
convert to decimal and percent
6/8
.75
75%
convert to decimal and percent
7/8
.875
87.5%
182
324
192
361
202
400
22
23
24
25
26
27
28
29
210
4
8
16
32
64
128
256
512
1024
32
33
34
9
27
81
42
43
44
16
64
256
52
53
54
25
125
625
172
289
