Unit Digit Patterns Flashcards
0
All powers of 0 end in 0
1
All powers of 1 end in 1
2
4 number pattern: 2-4-8-6
3
4 number pattern 3-9-7-1
4
2 number pattern 4-6
odd power = 4
even power = 6
5
All powers of 5 end in 5
6
All powers of 6 end in 6
7
4 number pattern 7-9-3-1
8
4 number pattern 8-4-2-6
9
2 number pattern 9-1
odd power = 9
even power = 1
When is a number divisible by 0?
No number is divisible by 0
When is a number divisible by 1?
All numbers are divisible by 1
When is a number divisible by 2?
A number is divisible by 2 if the ones digit is even.
When is a number divisible by 3?
A number is divisible by 3 if the sum of all its digits is divisible by 3.
When is a number divisible by 4?
A number is divisible by 4 if the last 2 digits are divisible by 4.
When is a number divisible by 5?
A number is divisible by 5 if the last digit ends in either 0 or 5.
When is a number divisible by 6?
A number is divisible by 6 if it is an even number whose digits sum to a multiple of 3.
When is a number divisible by 7?
Just do the division
When is a number divisible by 8?
A number is divisible by 8 if the number is even and the last 3 digits are divisible by 8.
When is a number divisible by 9?
A number is divisible by 9 when the sum of all the digits is divisible by 9.
When is a number divisible by 10?
If the ones digit is 0, then the number is divisible by 10.
When is a number divisible by 11?
A number is divisible by 11 if the sum of the odd number place digits minus the sum of the even numbered place digits is divisible by 11. For example
253 is divisible by 11 because 2+3 - 5 = 0
2915 is divisible by 11 because (9+5) - (2+1) = 11
When is a number divisible by 12?
When a number is divisible by both 3 and 4, it is divisible by 12
Odd + odd
even
even + even
even
even + odd
odd
odd - odd
even
even - even
even
even * even
even
even * odd
even
odd * odd
odd
even / odd
even
odd / odd
odd
even / even
even OR odd
What is the formula for division
x / y = Quotient + remainder / y
What are prime numbers less than 100?
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
How do you find the number of factors for a number?
- Find the prime factorization for the number
- For each factor, take the power + 1 and multiply
i.e. 240 = 2^4 * 3^1 * 5^1
number of factors = (4+1)(1+1)(1+1) = 20 factors
Given x y and x and y’s LCM and GCF
the LCM * GCF = xy
What do we know about prime factors of a perfect square
Other than 0 and 1, all its prime factors have to have even powers
What do we know about prime factors for a perfect cube
Other than 0 and 1, all of its prime factors have to have powers divisible by 3
What are some perfect squares?
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
What are some perfect cubes?
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000
What is the GCF of two consecutive integers
Two consecutive integers will NEVER share any prime factors. The GCF is always 1.
What is decimal of 1/6
0.166667
What is decimal of 1/7
0.142857
What is decimal of 1/9
0.1111111
What is the condition for determining if a decimal will terminate
A decimal will terminate if and only if the denominator of the reduced fraction contains a prime factorization of only 2s or 5s
If Z is divisible by X and Y, Z must also be divisible by
the LCM of X and Y, and any combination of factors up to the max of the LCM
If I square root an expression with a variable i.e. sqrt((x+3)^2) > 169
it becomes |x+3| > 13
Any fraction with a whole number numerator and nonzero whole number denominator is
either a terminating decimal or a repeating decimal.
How do you square decimals with 0s
Convert to scientific notation and then multiply accordingly, or just note that there will be 2*number of decimal places
What is square root of x^2
|x| not x!!!
What is square root of (x+y)^2
|x+y| not (x+y)!!
What is square root of 3
1.7
What is square root of 5
2.2
What is square root of 6
2.4
What is square root of 7
2.6
What is square root of 8
2.8
What is square root of 2
1.4
What is the same exponent base adding rule?
2^n + 2^n = 2^n+1
3^n + 3^n + 3^n = 3^n+1
so on so forth . . . .
What is the area of an equilateral triangle?
A = {side^2(root 3)} / 4
What is the longest segment of a rectangle
The diagonal sqrt(l^2 + w^2)
What is the maximum area of a rectangle with fixed perimeter
The maximum area of a rectangle is a square
What is the area of a hexagon
Area = (3 * sqrt(3)) / 2 * s^2
Described how inscribed angles in circles work
The inscribed angle will always be 1/2 the degree measure of the central angle
What does the right triangle inscribed in a circle suggest
The hypotenuse of a right triangle will be the circle diameter
If two lines are parallel
They will never cross each other
If an inequality has the < sign
you shade below the line
If an inequality has the > sign
you shade above the line
When you reflect a point (x, y) over the origin, it becomes
(-x, -y)
When you reflect a point (x, y) over the x axis, it becomes
(x, -y)
When you reflect a point (x, y) over the y axis, it becomes
(-x, y)
What is the formula for reflection of (x, y) over a point (a, b)
(2a - x, 2b - y)
How do we know if a graph can be the graph for a function?
Use the vertical line test - for any vertical line, there should be no more than one point intersection
In order to make any conclusions about terms in a sequence
The rule or formula must be known.
What is the arithmetic sequence formula
a_n = a_1 + (n-1) d where d is the common difference
What is the sum of the first n terms of an arithmetic sequence?
S_n = n/2 * (a_1 + a_n)
What is the geometric sequence formula
a_n = a_1 * r^n-1
Is 1 a prime number
1 is actually NOT a prime number because it does not have 2 unique factors!!
Percent means
per 100. Dividing a number by 100 is the equivalent of multiplying by 1/100
Translate the equations “p percent of q is equal to r”
p / 100 * q = r
What is the percent less than formula
Final Value = (Initial Value) * (1 - Percent Less than / 100)
What is the percent grater than formula
Final Value = (Initial Value) * (1 + Percent Greater Than / 100)
Ho do you represent markup and markdown on price
Given a percentage of x, multiply value by (1 + x/100) for mark up, and (1 - x/100) for mark down
How do you calculate precent change
Percent Change = [(Final Value - Initial Value) / Initial Value] * 100
Final Value = Initial Value * (1 +/- Percent Change / 100)
To return a discounted value back up to original value
We must increase the cost by some percent greater than the percent from which it was discounted
How many elements are in an inclusive set? (For example 50 to 101 inclusive)
Elements = Highest - Lowest + 1
101 - 50 + 1 = 52
How do you find the number of elements who are multiples of x between y and z
(highest # divisible by given number - lowest # divisible by given number) / given number + 1
How do you find the number of elements between two numbers (exclusive)?
Last - First - 1
How do you find the average of the elements in an evenly spaced set?
(first + last) / 2
How do we calculate the POSITION of median in an odd element set?
(number of elements + 1) / 2
How do we calculate the POSITION of the median in an even element set?
average between n / 2 and n / 2 + 1 terms
Does adding or subtracting the same amount from all terms change the standard deviation
Adding or subtracting the SAME amount from all terms does NOT change the standard deviation. We can have datasets with the same standard deviation but different averages.
Does multiplying or dividing the same amount by all terms change the standard deviation
Multiplying or dividing by the same amount will change the standard deviation.
How do you calculate standard deviation?
sqrt{[sum of (each points distance from mean)^2] / n + 1}
Standard deviation measures the average distance from the mean for a given set of points.
What is the basic combination formula
nCk = n! / (n - k)! k!
Explain the combination box and fill method
We have one box each time we have to make a decision and fill those with numbers. Then, we divide by the number of boxes factorial to make it a combination
If there are m ways to perform task 1 and n ways to perform task 2, how many ways are there to perform both tasks
m x n (assuming the tasks are independent)
When we are presented choices with the word “or” in combinatorics
We add instead of multiplying because the events are mutually exclusive
What does the word “at least” in a combinatorics problem imply?
“At least” typically implies the addition of outcomes
If x items from a group of y items must be chosen for a subgroup of z items
Mentally place those x items into the subgroup. Then y-x remaining items are being chosen for the z-x remaining spots in the subgroup.
Two events are collectively exhaustive if
together, those events represent all of the possible outcomes of a scenario.
If two events A and B are collectively exhaustive AND mutually exclusive
Total Number Of Ways Scenario Can Occur = Number of Ways in Which A Can Occur + Number of Ways in Which B Can Occur
What is the basic permutation formula
nPk = n! / (n-k)!
How do you solve a permutation that contains identical items
P = N! / (r1!) x (r2!) . . . where r1 and r2 are the respective frequencies of indistinguishable objects
What is the circular permutation formula
K items can be arranged in a circle in (k-1)! ways
What is the anchor method in permutations?
The anchor method asks for one to put the people/items in restricted positions in set spots first, then handle other items.
You have y items but x of them must be arranged together. How many ways can you arrange
(y-x+1)!(x)! Treat all items in a group as one unit.
How do you calculate probability of even A or B occurring when A and B are not mutually exclusive?
P(A or B) = P(A) + P(B) - P(A and B)
How do we know the number of leading zeroes in a decimal?
Write the decimal as a fraction of 1/x where x is an integer. There will be k-1 leading zeros where k is the number of digits unless x is a perfect power of 10, in which case there are k-2 zeros
What is the rate-time-work formula?
Rate x Time = Work
Rate = Work / Time
Time = Work / Rate
a/b > c/d if
ad > bc
Any factorial greater than 5!
Will have a trailing 0 as its units digit
How do you combine two ratios/proportions?
Find the LCM of the two values which connect the two ratios, then scale the proportions accordingly
If A and B are NOT independent, what’s the probability of A and B occurring?
P(A and B) = P(A) * P(B | A)
Given a rectangle with a fixed area, what is the minimum perimeter?
Minimum perimeter is given if the rectangle is a square.
