0
Q
23-101
A
-78
1
Q
An integer is divisible by 9 if
A
The sum of the integer’s digits is a multiple of 9
2
Q
What is the sum of the largest negative integer and the smallest positive integer?
A
0; -1 +1 =0
3
Q
An integer is divisible by 5 if
A
It ends in 0 or 5
4
Q
52/13
A
=4
5
Q
-6 x (-3 + (-5))
A
48
6
Q
1.25
A
1+ 2/10 + 5/100
6
Q
90/6=
A
15
7
Q
Rewrite 2/7 as a product
A
2 x 1/7
8
Q
Is zero odd or even?
A
Zero is an even integer
9
Q
Pi
A
Ratio of circle’s circumference to diameter
10
Q
(-2)^3 - 5^2 + (-4)^3=
A
-97
11
Q
✔A✔B=
A
✔AxB
12
Q
An integer is divisible by 2 if it is
A
Even
13
Q
Is 0 an integer?
A
Yes
14
Q
An integer is divisible by 3 if
A
The sum of the integer’s digits is a multiple of 3
15
Q
An integer is divisible by 10 if it
A
Ends in 0
16
Q
72/3
A
=24
17
Q
72/4
A
=18
18
Q
105/3=
A
35
19
Q
4? =104
A
?=26
20
Q
5•21=
A
105
21
Q
112/7=
A
16
22
Q
7•17=
A
119
23
90/18=
5
24
What is the first thing to do with a divisibility problem?
Prime factorization - factor tree
25
If x and y are both less than 5 what is the max value of xy?
Positive infinity
| Neg x neg = pos
26
What's the best starting point for divisibility problems?
Prime factorization to smallest numbers. Factor tree.
27
What is the factor foundation rule?
If a is divisible by b, and b is divisible by c, then a is divisible by c as well.
And if d is divisible by two diff primes e and f, then d is also divisible by e x f. ( if 20 is divisible by 2 and by 5 then 20 is also divisible by 2 x 5; 10)
28
Definition of least common multiple
Smallest number that is a shared multiple of both a and b
29
Why are LCM important in divisibility?
If x is divisible by a and b then x is divisible by the LCM of a and b. always.
30
What is the LCM of 2 numbers that don't share prime factors?
Their product.
LCM =3 x 10= 30
31
When 2 numbers share prime factors, what is their LCM?
It will be smaller than their product.
Don't double count evidence.
When you combine 2 factor trees if x that contrail overlapping primes, drop the overlap.
6 and 9 share prime factors so their LCM is not 6 x9= 54. In fact their LCM (18) is smaller than their product.
32
X is divisible by 6 and by 9. Is x divisible by 54, the product of 6 and 9?
Need LCM. Because x is divisible by LCM of 6 and 9, which is 18 (list multiples for both until find a match). So can only say mane divisible by 54. 6 and 9 share prime factors so LCM smaller than their product.
33
If integer a is not a multiple of 30, but ab is, what is the smallest possible value of integer b?
B=2
For integer a to be a multiple of 30, it would need to contain all of the prime factors of 30: 2,3,5. Since a is not a multiple if 30 must be missing at least one prime factor. B must supply missing factor. Smallest possible missing prime is 2. 2x15 =30
34
2^5
32
35
14^2
196
36
2^9
512
37
4^3
64
38
2^7
128
39
15^2
225
40
2^8
256
41
13^2
169
42
2^6
64
43
-3^2=
-(3^2) = -9; the negative of 3^2
Because of pemdas, square the 3 before multiplying it by negative 1 (-1). If wanted to square neg sign, put inside parentheses too, (-3)^2, which would be the square of neg 3.
44
Yx(Y^6)=
Y x Y^6 = Y^1 x Y^6 = y^1+6 =y^7
When multiplying exponential terms that have the same base, add the exponents. Treat any term without an exponent as if it had an exponent of 1. Also works with variables in exponent. 2^3 x 2^y = 2^3+y
45
(A^5)/(A^3)=
A^2
When dividing exponential terms that have the same base, subtract the exponents. Same with variables x^y/x^2= x^y-2.
46
A^5/A^5=
A^0= 1
For any non 0 value of a, we can say that a^0= 1
47
A^3/A^5 =
1/A^2 = a^(-2)
Value with a negative exponent is just one over that value with a positive exponent. It's the reciprocal. Number x it's reciprocal is always 1
48
2^(-3)
1/2^3= 1/8 = 0.125
49
5^(-3) x 5^(-6) =
5^(-3)+(-6) =5^(-9)
50
X^3/X^(-5)=
X^3 - (-5)= X^8
51
(5X^(-2))/y^3 =
5/((X^2)(Y^3))
Easier to make neg exponents positive. So if was neg up top becomes positive below. And visa-versa.
52
(-4)^(-3)=
1/(-4)^3 = 1/(-64) = -(1/64)
Odd powers of a negative base produce negative numbers. Also don't confuse the sign if the base with the sign if the exponent.
Note even powers of neg bases produce pos numbers. 1/(-6)^(-2)= (-6)^2=36
53
Negative power
One over a positive power
A^3/A^5= A^(-2)
A^(-2)= 1/A^2
54
1/(y^(-4))
= 1(y^4)/1 =y^4
If move entire denominator to switch sign of exponent, leave a 1 behind. And same goes if moving from numerator to denominator.
55
Simplify (a^2)^4
First square a. Then multiply the four a^2 terms together.
(A^2)(A^2)(A^2)(A^2)=a^(2+2+2+2)= a^8
Multiply the 2 and the 4 to get 8.
56
[x^(-3)(x^2)^4]/(x^5)
First simplify (x^2)^4= x^8
So fraction is now [x^(-3)(x^8)]/(x^5)
Then follow rules for multiplying and dividing terms with same base. Add and subtract exponents.
[x^(-3)(x^8)]/(x^5) = x^[(-3)+8-5]= x^0 =1
57
(2^2)(4^3)(16)=
Stop! Remember need same bases.
Try to break 4 and 16 down to prime factors.
4=2^2; 16=2^4
So (2^2)(4^3)(16)= (2^2)[(2^2)^3](2^4)=
(2^2)(2^6)(2^4)= 2^(2+6+4)= 2^(12)
58
(xy)^3=
(xy)(xy)(xy)=(x^3)(y^3)
When you apply an exponent to a product, apply the exponent to each factor.
59
(3/4)^(-2)
= (3^-2)/(4^-2)= (4^2)/(3^2)= 16/9
60
What is the prime factorization of 18^3?
Long way: 18x18x18
Instead do prime factorization of 18 to get 18= 2x9 = 2x (3^2)
18^3 = [2(3^3)]^3 = (2^3)(3^6)
61
(a^3)(b^3)=
(ab)^3
62
(2^4)(3^4)=
(2x3)^4= 6^4
63
13^5 + 13^3 =
Pull out common factor
= 13^3 x 13^2 + 13^3 =
(13^3)[(13^2) + 1]
64
(3^8)-(3^7)-(3^6)=
(3^6)[(3^2)-(3^1)-(3^0)] =
| 3^6)(9-3-1)= (3^6)(5
65
Add or subtract terms with the same base
Pull out a common factor
| (5^8) - (5^2) = (5^2)[(5^6) - 5^0]
66
Simplify
[(3^4)+(3^5)+(3^6)]/13
[(3^4)+(3^5)+(3^6)]/13=
Pull out common factor
{(3^4)[(3^0)+(3^1)+(3^2)]}/13 =
Simplify small powers
[(3^4)(1+3+9)]/13 =
[(3^4)(13)]/13 =
Cancel out 13s
=3^4
67
Add or subtract terms with different bases
Break down the bases and pull out a common factor.
| 2^3)+(6^3) = (2^3)(1+ 3^3
68
Add or subtract terms with the same base
Pull out the common factor
| 2^3)+(2^5)= 2^3(1+2^2
69
Add or subtract terms with different bases
Break down the bases and pull out the common factor
| 2^3)+(6^3)= 2^3(1+3^3
70
4^3
64
71
(9^x)^2=9
| What is x?
(9^2x)=9^1
Exponents must be equal. 2x=1, or x=1/2
For expressions with positive bases, a square root is equivalent to an exponent of 1/2
73
✔️7^22=
Take square root of positive number raised to a power
A) ✔️(7^22)= (7^22)^1/2= 7^(22/2)=7^11
B) ✔️7^22=✔️(7^11)(7^11)
An integer raised to a positive even power is always a perfect square.
For expressions with positive bases a square root is the same as an exponent if 1/2
74
If y^9 < y^4, describe all possible values
ANSWER: y < 1, but not equal to 0 (alternatively, 0 < y < 1 or y < 0).
Think about various categories of numbers: if y were negative, then y9 would also be negative, while y4 would be positive; then y9 < y4. If y = 0 or 1, then y9 = y4, which is not acceptable.
Test categories of numbers that tend to have different characteristics: negative integers, 0, fractions between zero and one, 1, and integers greater than 1.
If y = -1, then y9
76
How do you raise a number to a fractional power?
Apply 2 exponents -the numerator and the denominator as a root, in either order, ie
125^(2/3) = (3✔️125)^2 =5^2 =25
77
2^6
64
78
How many solutions does x=✔️25 have?
One, x=5
79
How many solutions are there when you need to take the square root of both sides of an equation? X^2 =25
Two, x=5 or x=-5
80
Ending pattern for multiples of 3
3,6,9,2,5,8,1,4,7,0
81
Pattern endings for multiples of 7
7,4,1,8,5,2,9,6,3,0
Gmat Math
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