Basic Flashcards
2^7=
128
3^3=
27
3^4=
81
5^3=
125
5/6 =
0.8333333333
5^4=
625
1/7 =
- 142857142857
0. 143
0^n =
0 if n > 0
7^3=
343
2^5=
32
2^3=
8
8^3=
512
1/8 =
0.125
2^4=
16
1/6 =
0.16666666667
3/8 =
.375
What is an irrational number?
Decimals that neither terminate nor repeat (e.g. pi, root 2, golden ratio). Cannot be written as integer over integer.
2^9=
512
5/8 =
.625
2^8=
256
7/8 =
.875
2^6=
64
1/9 =
.111111
1/20 =
.05
1/10 =
.1
1/100 =
.01
1/1000 =
0.001
1/600 =
=(1/6)(1/100)
=(1/6)(10^-2)
=0.001666667
4^4=
256
Volume of a cylinder
Pi x r^2 x h
4^3=
64
1^n =
1 for all n (even negative n)
6^3=
216
9^3=
729
What is a fraction x its reciprocal?
1
1 divided by any fraction =
The reciprocal of that fraction
If two fractions have the same numerator, but different denominators, which one is bigger?
The one with the smaller denominator. Bigger denominators make smaller fractions.
If the numerator gets bigger and the denominator gets smaller, what happens to the fraction?
It gets bigger
What happens if you add the same number to both the numerator and the denominator?
The resulting fraction is closer to one than was the original fraction
Suppose you add 2 to a numerator and 5 to a denominator. What is the new value of the fraction compared to the original fraction?
The new fraction is closer to 2/5 than the original fraction.
Sum of a set of integers =
Mean * number of integers
How do you find the lowest common multiple of a set of numbers?
E.g. 56, 7, 8
You prime factor each of the numbers and multiply the unique prime factors (if they are shared- take the one with the greatest exponent)
56: 2, 2, 2, 7
7: 7
8: 2, 2, 2
LCM: 2 x 2 x 2 x 7 = 56
Or you put the prime factors in exponent form and take all the unique prime factors and multiply them and also multiple the common prime factors with the largest exponent only.
24: 2^3 3^1
60: 2^2 5^1 3^1
LCM: 2^3 x 3^1 x 5^1
1/(a/b)=
b/a
When is a/b > c/d?
When ad > bc
Bow tie method
If a fraction is between 0-1, adding a positive constant to both the numerator and denominator will make the fraction bigger or smaller?
Bigger
If a fraction is more than 1, adding a positive constant to the numerator and denominator will make the fraction smaller or bigger?
Smaller
If a fraction is between 0-1, subtracting from the numerator and denominator a positive constant will make the fraction bigger or smaller?
Smaller as long as the new fraction is still positive.
If a fraction is more than 1, subtracting from the numerator and denominator a positive constant will make the fraction bigger or smaller?
Bigger as long as the new fraction is still positive.
2/7
.286
3/7
.429
4/7
.571
5/7
.714
6/7
.857
What can the units digit be of a perfect square?
Perfect squares cannot end in 2,3,7,8. The units digit will be 0,1,4,5,6, or 9
Most common pairs that result in powers of 10?
2x5
4x25
8x125
1!
=1
0!
=1
X^2 - 1 =
(X+1)(x-1)
(X^2 - 9)=
(X+3)(X-3)
4x^2 -100=
(2x+10)(2x-10)
X^2y^2 - 16 =
(xy + 4)(xy - 4)
(1/36)x^2 - 25 =
(1/6 x - 5)(1/6 x +5)
3^30 - 2^30 =
(3^15)^2 - (2^15)^2
=(3^15 + 2^15)(3^15 - 2^15)
(5!)^2 - (4!)^2
(5! + 4!)(5! - 4!)
10^4
10,000
Divisibility rule for 5:
Last digit needs to be 0 or 5
Divisibility rule for 4:
Last two digits form a 2 digit # divisible by 4.
AND if it ends in 00
Divisibility rule for 3:
Sum of digits need to be divisible by 3.
Divisibility rule for 9:
Sum of digits needs to be divisible by 9.
Divisibility rule for 6:
Needs to be divisible by 2 (even number) and divisible by 3 (sum of digits is divisible by 3).
What is the multiple and factor relationship?
If a number 1 is a factor of number 2, number 2 is a multiple of number 1.
Even/even
Odd or even
Even/odd
Even
Odd/odd
Odd
Odd/even
Not an integer!!
Even / 2
Remainder always 0
Odd / 2
Remainder always 1
When does multiplication result in an odd number?
Odd x odd
When does multiplication result in an even number?
Odd x even
Even x odd
Even x even
When do you get an even number when adding or subtracting?
Even +/- even
Odd +/- odd
How do you count the total amount of factors in a number?
Prime factorize. Then add one to all the powers of the prime factors and multiply them.
Eg- prime factors of 24 are 3^1 and 2^3. Add 1 to 1 and add 1 to 3 and then multiple them: 2x4=8 factors
When is the product of 2 positive integers the LCM?
When the 2 numbers don’t share any prime factors (eg 7&6 with 42)
How do you find the GCF of a set of positive integers?
Prime factorize, take common prime factors only (aka the one with the smallest exponent), and multiply them.
If there are no common prime factors, GCF is 1.
If x divides evenly into y and both x and y are positive integers, what is the LCM and GCF?
LCM = y GCF = x
X x Y = LCM (x y) x GCF (x y)
X x Y = LCM (x y) x GCF (x y)
The LCM of a set of positive integers provides us with…
All the unique prime factors of the set - thus is provides all the unique prime factors of the product of the numbers in the set.
The LCM can be used to determine when two processes that occur at different rates or times will coincide. Ex. Blinking light M flashes once every 12 seconds and Light L flushes every 32 seconds. If both lights flash together at 8:00pm, when will be the next time the lights will flash together again?
96 seconds later at 8:01:36
X is divisible by y, x/y=
Integer
X is a dividend of y, x/y =
Integer
X is a multiple of y, x/y=
Integer
Y divides into x (evenly), x/y =
Integer
Y is a divisor of x, x/y=
Integer
Y is a factor of x, x/y =
Integer
If x and y are positive integers and x/y is an integer, then
X/(any factor of y) is an integer
If z is divisible by both x and y, z must also be divisible of the LCM of x and y.
Eg. If z is divisible by 3 and 4,
It must also be divisible by 12.
Do not over-infer though. Just because z is divisible by 12, does not mean it’s divisible by 24.
Divisibility rule for 8:
If the number is even, divide the last 3 digits by 8- if there is no remainder, original number is divisible by 8.
AND all multiples of 1000 (ie numbers than end in 000) are divisible by 8 because 1000=125x8
Divisibility rule of 10:
If the ones digit ends in a 0
Divisibility rule of 11:
Ex. 253
Sum of odd numbered place digits - sum of even numbered place digits is divisible by 11.
253
=(3+2)-5=0, which is divisible by 11 as 0 is divisible by any number except for itself
Divisibility rule for 12:
Number needs to be divisible by both 3 and 4
What is the remainder formula?
X/y = Q + r/y
How to covert a decimal remainder into an exact integer? Eg. 1.8 = 9/5 : what is the integer version of .8?
Take the decimal portion (.8) of the result of the division and multiply it by the divisor (5) = 4
Can remainders be multiplied? Added? Subtracted?
Eg. Remainder when (12x13x17 / 5)
Yes for all:
For multiplication and addition: any excess remainders need to be counted for. EG. 2/5x3/5x2/5= 12/5 =2&2/5 so remainder is 2
For subtraction: correct for negative remainders by adding the denominator/divisor
What is the property of a remainder?
Non-negative integer that’s less than the divisor.
What must you know about any factorial >= 5!
It will always have 0 in its units digit
How are trailing zeros created in whole numbers?
By (5,2) pairs. Each pair creates 1 trailing zero. Thus the # of trailing zeros of a number is the number of (5,2) pairs in the prime factorization of that number).
Leading zeros: If x is an integer with k digits,
then 1/x will have k-1 leading zeros unless x is a perfect power of ten, in which case there will be k-2 leading zeros.
1/(5^5 x 2^5) => how many leading zeros?
There are 5 trailing zeros (5 pairs of 5&2) + 1 = 6 digits
Then apply the rule:
6 digits - 2 (since the denominator is a perfect square of 10) = 4 leading zeros
The product of any set of n consecutive positive integers is always divisible by all integers between 1 and n inclusive. Moreover the product of any set of n consecutive positive integers is divisible by n! In addition the product of any n consecutive integers must be divisible by all the factors of n!
What is the largest number that must be a factor of the product of any 4 consecutive integers?
n! = 4! = 24
The 4 consecutive integers must also be divisible by 4x3 or 4x2 or other factors of n!
What is the largest value of k such that 400!/5^k is an integer?
400/(5^1)= 80
400/(5^2)= 16
400/(5^3)= 3
= 99
If 90!/(15^n) is an integer, what is the largest possible value of the integer n?
90!/(15^n)
= 90!/(5x3)^n
5 is the limiting number (largest prime factor): 90/(5^1)=18 90/(5^2)=3 90/(5^3)=0 Great number of 15s in 90!= 21
All of the prime factors of a perfect square have…
All of the prime factors of a perfect cube have…
Even exponents
Exponents divisible by 3
When will a decimal equivalent of a fraction terminate?
When the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s or both.
When a whole # is divided by 10, the remainder will be….
Divided by 100, the remainder will be…
The units digit of the dividend
The last two digits of the dividend
When integers with the same units digit are divided by 5, what is the characteristic of the remainder?
the remainder will be constant (the same)
Will 2 consecutive integers ever share the same prime factors?
No, and their GCF will be 1
What is 1,000,000-456,789
=(1,000,000 - 1) - (456,789 - 1)
=999,999 - 456,788
=543,211
What is 999,999 + 456,789?
= 999,999 + 456,789
= (999,999 + 1) + (456,789 - 1)
= 1,000,000 + 456,788
= 1,456,788
How do you count the number of items in a set (inclusive)?
Eg. How many positive two digit integers are there?
Count = highest - lowest +1
=99 - 10 +1
=90
If n is an odd integer, then n^2 - 1 must be divisible by which of the following?
n^2 - 1 = (n+1) (n-1)
If n is odd, (n+1) and (n-1) must be even. (2 consecutive even integers).
The product of any x consecutive even integers will always be divisible by 2^x * x!
Must be divisible by 8.
0^0
1
