Arithmetic Flashcards
Number of Factors of an Integer
Every integer N is the product of powers of prime numbers:
N = p^αq^β· … · r^γ.
Number of integers :(α + 1)(β + 1) … (γ + 1)
Is zero an integer?
yes
name integers
-3,-2,-1,0,1,2,3…
even integer and formula
any integer divisible by 2 | can be expressed as 2N (N integer). -4, -2,0,2,4,6,8
Is Zero an even integer?
yes
odd integer and formula
not divisible by 2 | -3,-1,1,5,… | can be expressed by formula 2N+1 (N is an integer)
even X even = ?
even 2x4=8
even x odd = ?
even 2*3=6
odd x odd = ?
odd 3 x 5 = 15
even +/- even = ?
even 3+4=6
even +/- odd = ?
odd 2+3=5
odd +/- odd
even 3+5 =8
consecutive integers (provide examples)
are two or more integer written in sequence in which each integer is 1 more than the preceding one
prime number
positive integer that has exactly two different positive divisors: 1 and itself
first fifteen prime numbers
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47
what is digit?
digit refers to just one symbol, 9674 ; 9 - thousands digit, 6 - hundreds digit, tens digit, units digit
quotient
the result of division is quitient
square root of 2
1.4
square root of 3
1.7
a number is divisible
if the first number divided by the second number resulting in an integer
N/M=q what is N,M,q
m is a divisor of n, m is a factor of n, n is divisible by m, n is a multiple of m
a number is divisible by 2 IF
if the last digit is even! 2376 is div by 2
a number is div by 5 if
if the last digit is 0 or 5! 8745 or 8740
a number is div by 10 if
if the last digit is 0! 4500
a number is div by 4 if
if the number formed by last two digits is divisible by 4! 2376 -> 76 is div by 4
a number is divisible by 8 if
if the number formed by last three digits is divisible by 8!
2376 -> 376 is divisible by 8
a number is div by 3 if
the sum of all digits is div by 3
a number is div by 9 if
if the sum of all digits is div by 9 ! 2376 = 2+3+7+6=18 div by 9
a number is div by 12 if
if the number is div by both 3 and 4 ! 2376 div by 4, 2376 is div by 3
a number is div by 11 if
число делится на 11, если знакочередующаяся сумма его цифр делится на 11. 1234321. Знакочередующаяся сумма цифр этого числа равна 1 − 2 + 3 − 4 + 3 − 2 + 1 = 0. Так как 0 делится на 11, то и число 1234321 делится на 11
when a small number is div by large number integer then /?
the quotient is 0 and the reminder is the smaller integer: 5=7X0 + 5
Prime factors
positive integers are the prime numbers that divide that integer
prime factorization
every integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors
14=27, 81 = 3333
Least Common Multiple (LCM) - наименьшее общее кратное
Is the smaller positive integer that is a multiple of each of them
LCM of 14363 and 1950: use simple prime factorizaition
1. 14364 = 2^2 x 3^2 x 7 x 19
2. 1950 = 2 x 3 x 5^2 x 13
LCM = 2^2 x 3^3 x 5^2 x 7 x 13 x 19
Greates Common Factor, Divisor(GCF) - наибольший общий делитель
is the largest positive integer that is a factor of each of them. GCF 14363 and 1950 -prime factorization 1. 14364 = 2^2 x 3^2 x 7 x 19 2. 1950 = 2 x 3 x 5^2 x 13
GCF = 2 x 3
берем наименьшие степени чисел, которые есть в двух числах
Factorial of 0 = 0!
1
fractions
5/11 5 - numerator, 11 - denominator
reciprocal
means flip over a fraction: 2/5 -> 5/2
improper fraction
numerator is larger than or equal to their denominator 11/4
mixed fraction
have a whole number part and a fraction part
decimal fraction
fraction whose denominator is a power of ten 0.21
decimal point determines
determines the place value of the digits
98765.432: 9 - ten thousands, 8 - thousands, 7 - hundreds, 6 - tens, 5 - units. 4 - tenths, 3 - hundreds, 2-thousandths
ratio
shows the relationship between two o more numbers, but it does not tell you the actual values of those numbers
proportion and how to solve?
statement that two ratios are equal
Example: 2/3=8/12 in general X/Y=A/B
One way to solve is to cross multiply
Ratio and Proportions.
X+Y= 180
X : Y = 3: 7
Easies way:
x:y = 3:7 -> 10 parts
180/10 = 18 - value of 1 part
X=3*18
Percent; Whole 100% , part = p%. What percent?
p% = part/whole * 100. Example: 40 is what percent of 250? % = 40/250100=16%
250 is what percent of 40?
250/40100=625%
Percent; Whole 100% , part = p%. What is the value?
Part = p%/100% X Whole. Example:
What is 25% of 36?
25/100*36=9
Percent. What percent greater or less?
delta=Difference/whole * 100% Example: 150 is what percent greater than 120? (150-120)/120*100=25% 120 is what percent less than 150? (150-120)/150 * 100 = 20%
Percent. What percent increase or decrease?
Percent Increase = Amount of increase/original whole * 100
Percent decrease = Amount of decrease/original whole * 100
Example
If the price increases from $24 to $30, what is the percent increase in price?
(30-24)/24 X 100=25%
When dealing with percent decrease and increase always be careful ti put the amount of increase or decrease over the original whole. not the new whole!!!
Percent. What is the value …? Increase or decrease a number by certain percent.
To increase a number by r% multiply it by 100% + r%/100%
or (1+ r%/100)
Example: Increase a number by 15% ->
multiple (1+15%/100%) or 1.15
To decrease : multiply by (1+r/100)
Decrease a number by 15%
multiple by 0.85
Percent. An increase of a% followed by an increase of b%
always results in a larger increase than a single increase of (a+b)%
Example: An increase of 10% followed by an increase of 15%.
X * 1.1 * 1.15
Decrease the same but: X0.90.85
Average, arithemetic mean
Average = Sum of Values/Number of Values
Median
To calculate the median of n numbers. The list of numbers from the least to the greatest; if number of numbers is odd, the median is defined as the middle number. if n is even - average of the two middle numbers
mode
is the number that occurs most frequently in the list
{1,3,3,4,5,6} mode =6
{1,2,3,3,3,5,5,7,10,10,10,20} two modes = 3 and 10
range
difference between the largest and the smallest values
Standard Deviation
The more data are spread away from the mean the greater the standard deviation.
How to find the SD of n numbers:
- find the arithmetic mean
- find the difference between the mean and each of the n numbers
- square each of the differences
- find the average of the squared difference
- take nonnegative square root of this average
Example
{ 1, ,2,3,3,5,8,27}
1) mean = 7
2)difference:6,5,4,4,2,1,20
3)squares: 36,25,16,16,4,1,400
4) avg of squared = 71.7
5)square root(71.7) = 8.4
