Quant Essentials Flashcards
if a,b,c,d are positive, is a/b = c/d?
1) ad = bc
2) b = d
Equivalent fractions means that ad = bc
e.g. 1/8 = 7/56 -> 1x56 = 8x7
1 is sufficient, 2 is not.
Is 7/9 larger than 6/8?
Yes. Use bowtie method for positive fractions:
for a/b and c/d
if ad>bc
then a/b > c/d
Is 18/23 > 27/32?
No Find a common numerator and compare the denominator.
Larger denom -> smaller fraction
18/23 x 3/3 = 54/69
27/32 x 2/2 = 54/64
What are the DS choices?
A = First alone B = Second alone C = Both together D = Both alone E = Neither together
Split into AD + BCE blocks
1/6 as a decimal?
0.167
1/7 as a decimal?
0.143
1/9 as a decimal?
0.111
0.833 as a fraction?
5/6
When is x^2 < x < sqrt(x)?
When 0 < x < 1.
When x is between 0 and 1, the sqrt(x) is > x, and x^2 is < x
Perfect squares never end in?
2, 3, 7, 8
Because:
1, 4, 9, 25, 36, 49, 64, 81, 100
10011 - 7328 = ?
10011 = 9999 + 12
9999
- 7328
=2671 + 12 = 2683
What is 1/6 as a decimal?
0.167
5/6 = 0.833
What is 0.833 as a fraction?
5/6
0.167 = 1/6
What is 0.143 as a fraction?
1/7
2/7 = 0.286 6/7 = 0.875
What is 6/7 as a decimal?
0.857
5/7 = 0.714
What is 0.571 as a fraction?
4/7
5/7 = 0.714
3/7 as a fraction?
0.429
5/7 = 0.571
What are the positive 1-digit integers?
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
REMEMBER 0!
0! = ?
1
If x is not equal to y,
(x-y)/(y-x) =?
(x-y)/(y-x) = -1 (x-y)/(-1)(x-y) = -1
What are the 25 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 many different positive integers are factors of 12,000?
- Find the prime factorization
12,000 = 2^5 x 3^1 x 5^3 - Add 1 to the value of each exponent, then multiply.
Total number of factors = (5+1)(1+1)(3+1) = 48
If we know the LCM and GCF of two positive integers (x,y), we know their product.
x * y = LCM (x,y) x GCF (x,y)
e.g. If LCM = 24, GCF = 2,
then, xy = 24 * 2 = 48
Divisibility rule for 3
If sum of all digits is divisible by 3
Divisibility rule for 4
If last 2 digits are divisible by 4
Divisibility rule for 6
Even number where digits sum to multiple of 3
Divisibility rule for 8
Last 3 digits divisible by 8
Divisibility rule for 11
If sum of odd-place digits (1, 100, 10,000 etc) - sum of even-place digits (10, 1,000 etc) are divisible by 11.
e.g. 2,915 = (9+5)-(2+1) = 11, so 2,915 divisible by 11
Divisibility rule for 12
If divisible by 3 and 4
Product of n consecutive integers are divisible by
n!
e.g. product of 3 consecutive integers is divisible by 3!
Algebraic consecutive integers
- n^2 - n = n(n-1)
- n^2+n = n(n+1)
- n^3-n = n(n-1)(n+1)
- n^5-5n^5+4n = n(n+1)(n-1)(n+2)(n-2)
Consecutive even integers
If n is odd, both (n-1) and (n+1) are even.
(n-1)(n+1) = n^2 -1 = product of two consecutive even integers
Product of n consecutive even integers is divisible by
2^n x n!
e.g. 3 consecutive even integers visible by 2^3 x 3! = 8x6 = 48
Division, general formula
x/y = Q + r/y
Q = quotient, r = remainder
*manipulate the formula to solve for each of x, Q, r.
eg. Q = (x-r)/y
Any factorial >= 5! will
Always have 0 as its last digit, because it contains at least one 5 x 2 pair
Leading 0’s
are to the right of the decimal point (e.g. 0.02 has 1 leading zero, 0.78 has no leading zeroes)
If X is an integer with k digits and X is not a perfect power of 10, then 1/X will have how many leading zeros?
k - 1
e.g. 1/5000, X = 5000 (4 digits, not perfect power of 10), so decimal form 1/5000 has 3 leading zeros = 0.0002
If X is an integer with k digits and X is a perfect power of 10, then 1/X will have how many leading zeros?
k-2 leading 0s
e.g.
1/10 = 0.1 (0 leading 0s)
1/1000 = 0.001 (2 leading 0s)
Product of any set of n consecutive integers is divisible by…
all integers between 1 and n inclusive.
or, n!
e.g
5x6 is divisible by 2!
To determine the largest number of a prime number x that divides into y!
- divide y by x^1,…x^k, stopping when y/x^k produces a quotient of 0
- Sum quotients = number of prime number x in y!
What is the largest integer n such that 30!/4^n is an integer?
- 30!/4^n = 30!/2^2n
- use factorial divisibility shortcut, e.g. there are 2^15+7+3+1 = 26, 2^26 in 30!
- 2n <= 26, n=13
Largest value for n is 13
Reduced fractions with prime factorization with ONLY 2s, 5s or both will have a decimal that terminates.
Else, the fraction does not terminate.
e.g.
7/80 = 2^4 x 5^1, so it terminates
5/60 = 1/12 = 2^2 x 4^1, does not terminate
Units digit for powers for 2 and 8
2^n has pattern 2,4,6,8
8^n has pattern 8,4,2,6
Units digit for powers for 3 and 7
3^n has pattern 3,9,7,1
7^n has pattern 7,9,3,1
Which prime factors do two consecutive integers share?
None. Therefore, GCF(n, n+1) = 1
