Quant Essentials

Question 1

if a,b,c,d are positive, is a/b = c/d?

1) ad = bc
2) b = d

Answer 1

Equivalent fractions means that ad = bc
e.g. 1/8 = 7/56 -> 1x56 = 8x7
1 is sufficient, 2 is not.

Question 2

Is 7/9 larger than 6/8?

Answer 2

Yes. Use bowtie method for positive fractions:
for a/b and c/d
if ad>bc
then a/b > c/d

Question 3

Is 18/23 > 27/32?

Answer 3

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

Question 4

What are the DS choices?

Answer 4

A = First alone
B = Second alone
C = Both together
D = Both alone
E = Neither together

Split into AD + BCE blocks

Question 5

1/6 as a decimal?

Answer 5

0.167

Question 6

1/7 as a decimal?

Answer 6

0.143

Question 7

1/9 as a decimal?

Answer 7

0.111

Question 8

0.833 as a fraction?

Answer 8

5/6

Question 9

When is x^2 < x < sqrt(x)?

Answer 9

When 0 < x < 1.

When x is between 0 and 1, the sqrt(x) is > x, and x^2 is < x

Question 10

Perfect squares never end in?

Answer 10

2, 3, 7, 8

Because:
1, 4, 9, 25, 36, 49, 64, 81, 100

Question 11

10011 - 7328 = ?

Answer 11

10011 = 9999 + 12
9999
- 7328
=2671 + 12 = 2683

Question 12

What is 1/6 as a decimal?

Answer 12

0.167

5/6 = 0.833

Question 13

What is 0.833 as a fraction?

Answer 13

5/6

0.167 = 1/6

Question 14

What is 0.143 as a fraction?

Answer 14

1/7

2/7 = 0.286
6/7 = 0.875

Question 15

What is 6/7 as a decimal?

Answer 15

0.857

5/7 = 0.714

Question 16

What is 0.571 as a fraction?

Answer 16

4/7

5/7 = 0.714

Question 17

3/7 as a fraction?

Answer 17

0.429

5/7 = 0.571

Question 18

What are the positive 1-digit integers?

Answer 18

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

REMEMBER 0!

Question 19

0! = ?

Answer 19

1

Question 20

If x is not equal to y,

(x-y)/(y-x) =?

Answer 20

(x-y)/(y-x)       = -1
(x-y)/(-1)(x-y)   = -1

Question 21

What are the 25 prime numbers less than 100?

Answer 21

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

Question 22

How many different positive integers are factors of 12,000?

Answer 22

1. Find the prime factorization
   12,000 = 2^5 x 3^1 x 5^3
2. Add 1 to the value of each exponent, then multiply.
   Total number of factors = (5+1)(1+1)(3+1) = 48

Question 23

If we know the LCM and GCF of two positive integers (x,y), we know their product.

Answer 23

x * y = LCM (x,y) x GCF (x,y)
e.g. If LCM = 24, GCF = 2,
then, xy = 24 * 2 = 48

Question 24

Divisibility rule for 3

Answer 24

If sum of all digits is divisible by 3

Question 25

Divisibility rule for 4

Answer 25

If last 2 digits are divisible by 4

Question 26

Divisibility rule for 6

Answer 26

Even number where digits sum to multiple of 3

Question 27

Divisibility rule for 8

Answer 27

Last 3 digits divisible by 8

Question 28

Divisibility rule for 11

Answer 28

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

Question 29

Divisibility rule for 12

Answer 29

If divisible by 3 and 4

Question 30

Product of n consecutive integers are divisible by

Answer 30

n!

e.g. product of 3 consecutive integers is divisible by 3!

Question 31

Algebraic consecutive integers

Answer 31

1. n^2 - n = n(n-1)
2. n^2+n = n(n+1)
3. n^3-n = n(n-1)(n+1)
4. n^5-5n^5+4n = n(n+1)(n-1)(n+2)(n-2)

Question 32

Consecutive even integers

Answer 32

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

Question 33

Product of n consecutive even integers is divisible by

Answer 33

2^n x n!

e.g. 3 consecutive even integers visible by 2^3 x 3! = 8x6 = 48

Question 34

Division, general formula

Answer 34

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

Question 35

Any factorial >= 5! will

Answer 35

Always have 0 as its last digit, because it contains at least one 5 x 2 pair

Question 36

Leading 0’s

Answer 36

are to the right of the decimal point (e.g. 0.02 has 1 leading zero, 0.78 has no leading zeroes)

Question 37

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?

Answer 37

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

Question 38

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?

Answer 38

k-2 leading 0s
e.g.
1/10 = 0.1 (0 leading 0s)
1/1000 = 0.001 (2 leading 0s)

Question 39

Product of any set of n consecutive integers is divisible by…

Answer 39

all integers between 1 and n inclusive.
or, n!
e.g
5x6 is divisible by 2!

Question 40

To determine the largest number of a prime number x that divides into y!

Answer 40

1. divide y by x^1,…x^k, stopping when y/x^k produces a quotient of 0
2. Sum quotients = number of prime number x in y!

Question 41

What is the largest integer n such that 30!/4^n is an integer?

Answer 41

1. 30!/4^n = 30!/2^2n
2. use factorial divisibility shortcut, e.g. there are 2^15+7+3+1 = 26, 2^26 in 30!
3. 2n <= 26, n=13

Largest value for n is 13

Question 42

Reduced fractions with prime factorization with ONLY 2s, 5s or both will have a decimal that terminates.

Answer 42

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

Question 43

Units digit for powers for 2 and 8

Answer 43

2^n has pattern 2,4,6,8

8^n has pattern 8,4,2,6

Question 44

Units digit for powers for 3 and 7

Answer 44

3^n has pattern 3,9,7,1

7^n has pattern 7,9,3,1

Question 45

Which prime factors do two consecutive integers share?

Answer 45

None. Therefore, GCF(n, n+1) = 1