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Exponents
1) Definition
a 1
0
a
n
1
 n
a
n
m
a  m an
2) Properties
a n  a m  a nm
a n  a m  a nm
(a n )m  a nm
(ab)n  a nbn
1. If 24 x  24 x  24 x  24 x  425 , What is x?
Answer: 2+4x=50, x=12
2. If 42 x  2  163 x 1 , what is the value of x?
* If 27 2 x  4  33 x 9 , then x=?
Answer: 6x+12=3x+9, x=-1
7
* If (5k ) 3  25 , then k=?
x
y
x
y
3. If (2 )(2 )  8 and (9 )(3 )  81 then what is 2x-y?
4. What is the smallest integer n for which 25n  512 ?
Answer: n=7
5. If a is a positive integer greater than 1, then 2a  2a1 
(a) 3a 1
(b) 2a1
(c) 2a
(d) 2a a 1 (e) 3(2a )
Answer: e
6. The value of
214  215  216  217
17
is how many times the value of 2 ?
5
5
8
7
6
* 3 3 3 3 =
Answer: 35 (1  33  32  3)  16  35
7.
x 5 ( x 4 )2 x3
?
x( x 5 )4 x 7
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Remainders
If you divide a  b , then you will get quotient : q and remainder : r .
In this case, you can rewrite a number a by b, q, and r
a  bq  r
ex) If you divide 26 by 3, then you will get the quotient 8 and remainder 2.
So…. you can rewrite 26=3*8+2
1. When m is divided by 5, the remainder is 2. What is the remainder when 3m is
divided by 5?
Answer: 1
2. When positive integer x is divided by positive integer y, the remainder is 9. If
x
 96.12 , what is the value of y?
y
Answer: 75
Solution:
yk  9
 96.12
y
9
 75
0.12
3. What is the remainder when the integer x  9 n is divided by 10?
(1) n is a positive even integer.
(2) n is a multiple of 4
Hint) Unit digit
Answer: D
9,1,9,1,9,1…
Therefore, k  96 and y 
4. What is the tens digit of positive integer x?
(1) x divided by 100 has a remainder of 30
(2) x divided by 110 has a remainder of 30
Answer: A
5. What is the remainder when the positive integer x is divided by 3?
(1) x is an even integer larger than 2
(2) x is a multiple of 9
Answer: B
6. If positive integer x is divided by 2, the remainder is 1. What is the remainder when x
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is divided by 4?
(1) 31< x<35
(2) x is a multiple of 3
Answer: A
Sol) x is an odd number
x=3(2k+1)=6k+3=(4+2)k+3=4k +2k+3
if k=2i+1, remainder is 1
if k=2i, remainder is 3
7. If n is an integer greater than 6, which of the following must be divisible by 3?
Hint: pick any number satisfying the condition.
(A) n(n+1)(n-4)
(B) n(n+2)(n-1)
(C) n(n+3)(n-5)
(D) n(n+4)(n-2)
(E) n(n+5)(n-6)
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Multiple and Factor
C=A*B
C is the multiple of A and B
A is the factor of C
B is the factor of C
1. Is the integer y a multiple of 4?
(1) 3y 2 is a multiple of 18
p
(2) y  , where p is a multiple of 12 and q is a multiple of 3.
q
Answer: E
2. What is the greatest positive integer x such that 3x is a factor of 910 ?
3. If x and y are nonzero integers and 450x=120y then which of the following must be
an integer?
xy
i.
60
15 x
ii.
4y
4x
iii.
15 y
Answer:
xy 15 x
60 , 4 y
Hint: Factorize each number to analyze the problem.
Sol) 15 x  4 y  3  5 x  2  2 y
4. If a certain number is divisible by 12 and 10, it is not necessarily divisible by which of
the following?
(A) 4
(B) 6
(C) 15
(D) 20
(E) 24
Answer: E
Hint: Factorize 12 and 10
5. If p, q and r are positive integers such that q is a factor of r,
and r is a multiple of p,
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which of the following must be an integer?
a)
b)
c)
d)
e)
pq
r
rq
p
p
q
pq
r
r ( p  q)
pq
Answer: E
hint: r is the multiple of p and q.
2x
6. If x and y are positive integers, is y
an integer?
(1) Some factors of y are also factor of x
(2) All distinct prime factors of y are also prime factors of x
Answer: E
7. If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a
multiple of 105?
(1) x is a multiple of 9
(2) y is a multiple of 25
Answer: B
Hint: Factorize each number to analyze the problem.
Sol) x=2*3*l
y=2*7*m
xy=3*5*7*n
8. If 6x=8y=14z, then what is a possible sum of positive integers x, y and z?
(A) 52
(B) 58
(C) 84
(D) 122
(E) 168
Sol)
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6x=8y=14z
 2  3 x = 2  2y 2  z2 7
Therefore,
x  22  7  k
y  3 7  k
z  22  3  k
x  y  z  28k  21k  12k  61k
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Square Root
1. Definition
x 2  a  x   a (a : real number )
2. Property
 a
2
a
a 2 | a |
In many case, this property is combined with x 2  y 2  ( x  y)( x  y)
Ex) ( 3  7)( 3  7)  ?
1. What is x 2 y 2 if x<0 and y>0?
a) -xy
b) xy
c) -|xy|
d) |y|x
e) No solution
Sol)
x 2 y 2  ( xy ) 2 | xy |
2. What is
98  32  8 ?
3. ( 2  1)( 2  1)( 3  1)( 3  1)  ?
4.
22 3
10  300
2
5. If 2  d  5 , what is the value of (d  3)
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Unit Digit
Find the pattern.
Do you know how to find the pattern?
39
1. What is the unit digit of 2 ?
19
15
2. What is the unit digit of 9  7 ?
3. What is the unit digit of?
(23)6 (17)3 (61)9
30
4. What is the sum of all digits for the number 10  37 ?
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Quadratic Equation
ax 2  bx  c  0
1. Solving by factorization ax 2  bx  c  0  a( x  p)( x  q)  0  x  p or q
b
c
2. sum of two roots: 
product of two roots:
a
a
You don’t need to memorize the general formula because mathematics doesn’t
focus on the memorization but understanding.
2
1. If 3 is one solution of the equation x  2 x  k  9 , where k is a constant, what is the
other solution?
a.
b.
c.
d.
e.
-2
-1
0
1
2
2
2. Which of the following equations has a root in common with x  7 x  12  0 ?
a. x 2  1  0
b. x 2  x  12  0
c. x 2  8 x  4  0
d. 3x 2  9  0
e. x 2  x  6  0
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Decimal, Fraction
Just watch how numbers are moving to.
1 4
6

Ex)
4 6
3 3
4
4
,
Some useful tips.
You can switch x with number 2 so that you can solve the rational equation quickly.
3
3
2 x
x
2
Don’t try to calculate first. Instead, reduce the number first.
3 2 3

4
2
Fraction or decimal is a good material to make a problem difficult that’s why you can see
the fraction and decimal problems in the test a lot.
15
 48
2
?
3
0.0024  5 
4
0.025 
1.
2.
1
1
1

0.03 0.37
3. if s 
?
2
n
1 2

x 3x
, nx  0 , what is the value of s?
(1) x=2n
(2)n=1/2
Answer: A
2
n
2
 3nx
6x
6x
n
Sol) s 



1 2
1 2

(  )  3nx 3n  2n 5n
x 3x
x 3x
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Ratio, Percentage
Essentially, those two topics are same. Here are the expressions.
1
The ratio 1 to 4  1: 4   .25  25%
4
Need to know two topics about percentage.
1. Percentage of what
Ex) 40% of 40 can be expressed by 0.4*40
2. Percentage increase (or decreage)
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Inequality
1. Is x>y?
(1) x 2  y 2
(2) x  y  0
Answer: B
2. Is x 2  y 2 a positive number?
(1) x- y is a positive number
(2) x+y is a positive number
Answer: C
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