Quant Flashcards
In data sufficiency questions, what are we being asked to calculate?
NOT the numerical answer e.g., x=5, but rather, could you get the value of x? so only analyse up until the point where you can say yes or no!
what are the 5 strategic numbers to evaluate statements?
- positvie integer
- positive fraction (0-1)
- 0
- negative integer
- negative fraction (-1 -0)
simple fractions. what is a proper and improper fraction?
proper fraction: numerator is less than the denomenator (5/7)
improper fraction: numerator is greater than the denomenator around (3/2)
what is mixed number?
number + proper fraction (7/2 = 3 1/2, 2 is called quotient)
Least Common Denominator (LCD) is… ?
LCD of 1/2, 1/3, and 1/8 be..?
smallest non-zero whole number that is divisible by each of the denominators
24
what is Equivalent fraction?
Same value but appear different a/b = c/d but also ad=bc
Rational number, integer, whole number, natural number?
Rational: any numbers incl. fraction
integer: incl. 0 and negatives
whole number: 0, 1,2,3,4…
Natural number: 1,2,3,4,…
What is reciprocal of 5? what is the only number that doesn’t have reciprocal?
1/5, 0
complex fraction
fractions that numerator, denominator or both are also fractions
compare the size of 4/17, 8/49 and 12/75?
24/102, 24/147 and 24/150. So compare using numerator.
what is the units digit?
same as ones, units digit for 675 is 5
whats the tenth and hundredth of 7.54952?
tenth: 5, hundredth: 4
What is principal square root?
The unique nonnegative square root of a nonnegative real number. principal square root of 121 is 11 although root of 121 is +/- 11.
- real number includes rational and irrational number like root 2.
- we only consider principal square root when √ is used.
x^2 < x < √x under what condition of x?
0
what is a perfect square?
A number made by squaring a whole number.
What is the opposite of the number 7?
-7
Whats the order of math calculations?
P: Parentheses or absolute value bars or radicals
E: exponets
M and D: multiple and division from the left
A and S: addition and subtraction form the left
what is 0!
1
List 2 properties of 0 involving multiples and factors.
0 is a multiple of all numbers.
0 is the only factor of itself.
List 3 properties of 1 involving multiples, factors, and prime numbers.
every number is a multiple of 1.
1 is a factor of all numbers.
1 is not a prime number.
Rules of addition, multiplication and division between odd and even numbers?
Addition: only combination of odd and even leads to odd numbers.
Multiplication: only odd times odd will be odd.
Division: even divided by odd will be even.
odd divided by odd will be odd.
even divided by even can be either.
What is the definition of prime number?
Any integer greater than 1 and has no factors other than 1 and itself
What are composite numbers?
Number that is not a prime number
How do you calculate the number of factors does the value 12,000 has?
- prime factorize 12,000 = 5^3 * 3 * 2^5
- add 1 to each of the exponents
- multiply all of the the (exponent + 1)s
(3+1)2(5+1)=48 factors.
How many prime factors and unique prime factors do 45 have?
3^2* 5 so you have 3 prime factors and 2 unique prime factors.
How many unique prime factors do the prime numbers have?
One. Prime numbers have 2 factors, 1 and the number itself. Since 1 is not the prime number, the number itself is the only unique prime factor.
What is the range of numbers when I say numbers between 1 and 10?
2-9
Does a number of unique prime factors of x change when you squared the x?
No.
6 = 23 –> 2 UPFs
36 = 2^23^3 –> 2 UPFs
What is LCM?
The smallest positive multiple of all integers in the set.
Careful with the 3 or more variables in the set!
What is the GCF?
The positive integer that the largest number that divides into all of the numbers in the set
If we know LCM(x,y) and GCF(x,y), what can we calculate?
xy
What does x is divisible by y mean?
x/y has no remainder
and x is not divisible if (5√x)/5 = √x as x could be integer or not
If integer x is divisible by 15 and 20, does x also be divisible by LCM(15,20)?
Yes
Divisibility rule: How do you know if the number is divisible by 4?
Last 2 digits are divisible by 4.
For 244, 44/4=11 so its divisible.
00s are divisible by 4
Divisibility rule: How do you know if the number is divisible by 6?
Even number + sum of all digits is divisible by 3
Divisibility rule: How do you know if the number is divisible by 8 ?
even + last 3 digits is divisible by 8
000s are divisible by 8.
Divisibility rule: How do you know if the number is divisible by 9?
Sum of all digits is divisible by 9
Divisibility rule: How do you know if the number is divisible by 11?
Sum of odd digits - sum of even digits is divisible by 11.
for 2915, 5+9 - (1+2)=11
for 253, 2+3-5=0 so divisible.
Divisibility rule: How do you know if the number is divisible by 12?
Last 2 digits is divisible by 4 + sum of the digits divisible by 3.
The product of n consecutive products are divisible by…?
n!
and if x is 3 consecutive products and y is 4 consecutive products, the xy is divisible by 3!*4!
THe product of n consecutive even integers is divisible by…?
(2^n)*n!
what is the equation for x divides into y?
and x divided by y?
into: y/x
by: x/y
What is the remainder when x=131417 is divided by 5?
- divide indivisually
13/5 = 2R3
14/5 = 2R4
17/5 = 3R2 - multiply all the reminders
342=24 - Take the reminders if multiply is excessive reminders
24/5=4R4 - So 4 quotients can be made from sum of remainders and what is left (remainder of the remainder) will be the overall remainder.
What is the remainder for x= 17-13?
- divide indivisually
17/5=3R2
13/5=2R3 - subtract (add if the x is addition equation) the remainders
2-3 = -1 - Remainder -1 does not make sense. If you look at the equation, 3R2 - 2R3 = 1R-1. Thus, break out the quotient of 1 and it will be 0R4. Thus remaineder is 4.
※This technique of quotient reduction idea can only be used on adding and subtracting.
prime factorization of (5*2) creates what?
Trailiing 0
How do you find how many digits in a particular number?
- prime factorize
- count (5*2) pairs
- Multiply all rests
- add digits of 3 and 2
What is leading zero? and what are the 2 rules ?
Leading 0 is number of 0s after decimal point.
Denomenator is a perfect power of 10: K(number of denomenator digits) -2
Denomenator is NOT a perfect power of 10: K -1
400! = (15^n)X what is the max number of n?
Finding number of prime numbers in the factorial
15^n = 3^n * 5^n so that since 5 appears less than 3, you only need to calculate how many 5s in the 400!.
To do that, 1. Divide 400 by 5, 5^2, 5^3... until Quotient is 0 400/5 = 80 400/25 = 16 400/125 = 3 (ignore the R)...
- add all the Quotient
80+16+3+… = 99
Since you can divide 400! by 3 more times, there will be 99 sets of 15 in 400!
How many times can 400! be divided by the 45?
400!/ 45^n = k(integer)
45^n = 3^2n * 5^n thus we dont know which comes around more.
- you calculate how many times you can divide 400! by 3 and find 196.
- since we wanna know how many 9s in 400!, there will be 196/2 = 98 9s.
- we know there will be 99 5s can fit in the 400! and thus, we know you can divide 400! by 45 98 times.
What is perfet square?
if √x = integer, than x is perfect square
What is the rule of the prime factorizaion of perfect square?
Will always contain even exponents 100 = 2^2 * 5^2
Dicimal will terminate iff …?
The denomenator of the equivalent fraction calue has prime factorization that only contain 2s or 5s or both. Otherwise, it will not terminate.
What is the remainder of 3^320 / 4?
Remiander repeats! 3^1: 3/4 = 0R3 3^2: 9/4 = 2R1 3^3: 27/4 = 6R3 3^4: 81/4 = 20R1
3,1,3,1,… sequence so that 320/4=80 so even number will have R1.
What is a value fo units digit for 7^49?
1. Find the pattern in units digit 7^1 = 7 7^2 = 49 7^3 = 343 7^4 = ~~1 7^5 = ~~7
So the sequence of units digit of 7 is 7,9,3,1.
- Find where the unit digit of 49th number of sequence is.
49/4 = 12R1 thus the unit digit must be 7.
What are the rules regarding the remainder with the positive integers with the same unit digit when divided by 5?
remainder is constant. 7/5 = 1R2 17/5 = 3R2 57/5 = 11R2 when finished with 7, the remainder is always 2 in the specia case that same units digits and divided by 5 only!
2 consecutive integers will never…?
Will never share the same prime factors.
i.e., GCF(n, n+1) = 1
Explain the rule of the n roots when n is even and odd.
n is even: n√(x^n) = |x|
n is odd: n√(x^n) = x and x can be negative
simplify 4/(a - √(b)
Take conjugate of the binomial (a - √(b)), which is a + √(b) and multiply to both denomenator and numerator.
4(a + √(b))/ (a^2 - b)
What do you have to watch out for whn ou soling a equation involving the square roots?
Make sure after you get the x, you put them back into the equation to see if its correct.
if n is even in n√a, a must be?
greater than equal to 0. As there are no number such that
x^n = negative number. (-4)^2 = 16
x^a * y^a is ?
(xy)^a as long as exponents are the same, you can multiply and divide them together
