Quant Ch 3 - Properties of Numbers Flashcards
Integer
A number with out a decimal or fractional component
x⁰
= 1 when x ≠ 0
The first prime number
- 1 is not prime
Representation of even integers
2n
Representation of odd integers
2n-1 or 2n+1
Addition rules for even and odd numbers
O+O = E
E+E = E
O+E = O
E+O = O
Subtraction rules for even and odd numbers
O-O = E
E-E = E
O-E = O
E-O =O
Multiplication rules for even and odd numbers
ExE = E
ExO = E
OxO = O
Division rules for even and odd numbers
E/O = E
O/O = O
E/E = O or E
Only apply when 1 integer divides evenly into another integer
Factors
Numbers that divide into a larger number evenly
Multiples
The product of a number with any integer
Prime numbers
Numbers that have no other factors other than 1 and itself
Composite number
Any number that is not a prime number
Prime factorization
When a composite number is broken down and expressed as the product of its prime factors
Finding the number of factors of an integer
1) find the prime factorization of the number
2) Add 1 to the value of each exponent & multiply the results
Unique prime factors
The number of prime factors that are different in a prime factorization. The number of unique prime factors in a number does not change when the number is raised to a positive integer exponent
Least common multiple (LCM)
Smallest positive integer into which all of the numbers in a set will divide
Finding the LCM
1) find the prime factorization
2) of any repeated prime factors, take the highest exponent once
3) take all non-repeated prime factors of the integers
4) multiply what’s found in steps 2-3. the result is the LCM
In any set of (+) integers, the LCM is ≥ the largest number in the set
Repeated prime factors
A prime factor is repeated when that prime factor is shared by at least 2 of the numbers in a set
Integers with no common prime factors
If a set of positive integers share no prime factors, the LCM is the product of the set of numbers
Greatest common factor (GCF)
Largest number that will divide into all numbers in a set
Find the GCF
1) find the prime factorization
2) of any repeated prime factors, use the factors with the smallest exponent
3) multiply the factors in step 2
If there are no prime factors in common, the GCF is 1
In any set of (+) integers, the GCF is ≤ the smallest number in a set
LCM and GCF when 1 integer divides into another evenly
Given 2 positive integers x and y, if it is known that y divides into x evenly then:
LCM (x,y) = x
GCF (x,y) = y
xy = LCM(x,y) x GCF(x,y)
How to use the LCM to find unique prime factors
If we know the LCM of a set of (+) integers, the prime factors of the LCM = the unique prime factors of the whole set
How to use the LCM for repeated pattern questions
When trying to determine when 2 processes that occur at differing rates or times will coincide, the answer is the LCM
*unique question
Even division
When the numerator of a (+) fraction is a multiple of the denominator (and vice versa)
How to approach divisibility questions
Easiest way to determine whether x is a multiple of y is to prime factorize x and y
Factors of factors rule
In x/y, if x is divisible by y, then x is also divisible by all factors of y
Divisibility with exponents
x^a / x^b = x^a-b
Formula for division
x/y = Q + r/y
x = Qy +r
Q = x-r/y
r = x-Qy
Multiplying remainders rule
When determining the remainder of x and x is the product of positive integers, the remainder of x is the product of the remainders of each positive integer
*do a practice problem on this if it doesn’t make sense
Determining the number of trailing zeros of a number
If the prime factorization of a number contains a (5x2) pair, that pair corresponds to 1 trailing zero in the number
Any factorial ≥ 5 will always have zero as its units digit
Leading zeros in decimals
0.2 = no leading zeros
0.02 = 1 leading zero
Determining the number of leading zeros in a decimal
If x is an integer with k digits, and if x is not a perfect power of 10, then 1/x will have k-1 leading zeros
Perfect squares
When the square root of an integer x is and integer, x is a perfect square
All perfect squares end in…
0,1,4,5,6, or 9
The prime factorization of a perfect square will contain only
even exponents
Perfect cube
A number (not 0 or 1) that all prime factors have exponents that are divisible by 3
Terminating decimals
Decimals that have a finite number of digits
Determining whether a decimal is terminating
Any fraction with a denominator whose prime factorization contains ONLY 2s or 5s or both, produces decimals that terminate
Remainders after division by 10
When an integer is divided by 10, the remainder will be the units digit of the numerator
Remainders after division by 5
When integers in a set with the same units digit are divided by 5, the remainder is constant between them
