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Manhattan Prep GMAT Number Properties
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Terms in this set (145)
Interger
positive and negative numbers that are whole. Includes zero
Arithmetic Processes that ALWAYS result in Interger
addition
subtraction
multiplication
Arithmetic Process that SOMETIMES results in Interger
division
Intergers are Divisible by 2 if
the interger is even
Intergers are divisible by 3 if
the sum of the interger's digits is divisible by 3
Intergers are divisible by 4 if
the interger is divisible by 2 twice or if
the last two digits are divisible by 4
Intergers are divisible by 5 if
the interger ends in 0 or 5.
Intergers are divisible by 6 if
the interger is divisible by both 2 and 3.
Intergers are divisible by 8 if
the interger is divisible by 2 three times or
if the last three digits are divisible by 8.
Intergers are divisible by 9 if
the sum of the interger's digits are divisible by 9
Intergers are divisible by 10 if
the interger ends in 0.
Factor
also known as a divisor.
divides evenly into an interger.
1,2,4 are all factors of 8. 2,3 are factors of 6.
Multiple
the result of multiplying an interger by any other interger.
8,16,24 are multiples of 8. 12,24,36 are multiples of 6.
An interger is always a ____ and ____ of itself
factor and product
GMAT Mindfulnesss: Divisibility
GMAT will say that x is divisible by y in several different ways. Learn this phrasing and mentally convert them into a single form when you see them.
If you add or subtract multiples of (N),
then the result is a multiple of (N).
If (N) is a divisor of x and of y,
then (N) is a divisor of x+y
Prime Number
any positive interger larger than one with exactly two factors: one and itself.
What is the only even prime number?
2
Give the first ten prime numbers
2,3,5,7,11,13,17,19,23,29
Prime factorization
Very helpful way to analyze a number. Prime factors of a number can determine all factors of a number.
Factor Foundation Rule
If (a) is a factor of (b), and (b) is a factor of (c) then (a) is a factor of (c).
Odd ± Even
Odd
Odd ± Odd
Even
Even ± Even
Even
Odd x Odd
Odd
Even x Even
Even
Odd x Even
Even
Will Even / Even result in Even?
Yes
Will Even /Even result in Odd?
Yes
Will Even/Even result in non-interger?
Yes
Will Even / Odd result in Even?
Yes
Will Even/ Odd result in Odd?
No
Will Even/ Odd result in non-interger?
Yes
Will Odd/ Even result in Even?
No
Will Odd/Even result in Odd?
No
Will Odd/Even result in non-interger?
Yes
Will Odd/ Odd result in Even?
No
Will Odd/ Odd result in Odd?
Yes
Will Odd/ Odd result in non-interger?
Yes
Sum of All Primes
will always be even, unless one of the primes is two.
Because all primes,except two are odd.
Evenly Spaced Sets
sequences of numbers whose values go up and down by the same amount (increment) from one item in the sequence to the next.
Consecutive Multiples
all of the values in te set are multiples of the increment.
Consecutive Intergers
all values increase by 1, all intergers are multiples of 1
Consecutive Intergers are a subgroup of what?
consecutive multiples and evenly spaced sets.
consecutive inter.<consecutive multiples<Even.Spaced sets
What makes an even spaced set fully defined?
(1) the smallest (1st) or largest (last) number in the set
(2)The increment (always 1 for consecutive intergers)
(3)The number of items in the set
What is the average in an evenly spaced set equal to?
median
How do you calculate the median of an evenly spaced set?
find the average of the first and last terms.
(first + last)/2
How do you calculate the sum of the elements in an evenly spaced set?
average x the number of items in the set
How do you calculate the number of items in a set of consecutive intergers?
(last-first) +1
How do you calculate the number of items in a set of consecutive multiples?
[(last -first) /increment ] +1
The average of a set of consecutive intergers with an ODD number of items
will ALWAYS be an interger
The average of a set of consecutive intergers with an EVEN number of items
will always be a non-interger.
Is the product of (n) consecutive intergers divisible by n! ?
Yes
If (n) is odd, then is the sum of (n) consecutive intergers divisible by (n)?
Yes
Result is always interger
If (n) is even, then is the sum of (n) consecutive intergers divisible by (n)?
No
Result is always non-interger
Why is the statement "the product of (k) consecutive intergers will always be divisible by k!" a true statement?
According to the factor foundation rule, every number is divisible by all the factors of its factors. Thus the product of (n) consecutive intergers is divisible by (n-1), (n-2), (n-3), (n-4)....; because the set of intergers has multiples of all these numbers within it.
When the number of items in a set of consecutive intergers is odd, what can be said about the sum of the interger?
It is always a multiple of the number of items.
When the number of items in a set of consecutive intergers is even, what can be said about the sum of the interger?
It is never a multiple of the number of items.
What will be the sign of an even exponent?
even exponents can be positive or negative.
What will be the sign of an odd exponent?
odd exponents maintain the sign of the base.
0^x
always 0
1^x
always 1
-1^even
1
-1^odd
-1
What if you see x^6=x^7=x^15 on the GMAT?
The answer is either 0 or 1, because of the odd exponents.
What if you see x^6=x^8=x^10 on the GMAT?
The answer is would be 0,1, or -1 , because even exponents can be negative of positive.
When a base is between 0-1 (fraction/decimal), what happens as the exponent increases.
The value decreases, because fractions multiplied by fraction move closer to zero (become smaller).
What should you do when you see a base without an exponent?
Write an exponent of 1.
What is the value of any non-zero number when raised to 0?
1
How do exponential fractions tell us to manipulate numbers?
Numerator-> what power to raise base to
Denominator-> what root to take
X^a * X^b
X^a+b
a^x * b^x
(ab)^x
x^a/x^b
x^a-b
(a/b)^x
a^x/b^x
(a^x)^y
a^xy
x^-a
1/x^a
x^a/b
b root of x raised to the a
What are the possible values of even roots?
Even roots only have positive values
What are the possible values of odd roots?
Odd roots will have the same sign as the base
When are roots negative?
When the root is odd and the base is negative
If a root is even, then will the base be negative?
No, it is impossible to have an even root with a negative base
What is the conversion point between exponents and roots?
fractional exponents
True or False:
When simplifying roots outside of the radical, you can combine them with multiplication and division
True
True or false:
When simplifying outside of the radical, you can combine them with addition and subtraction
False
What is a good way to simplify imperfect roots?
prime factorization
When estimating imperfect squares without coefficient what strategy should you use?
find out two closest perfect squares on either side of the number.
When estimating imperfect squares with coefficient what strategy should you use?
Combine coefficient with the root by placing it back inside of the radical, then find out two closest perfect squares on either side of the number
n root of x/ n root of y
n root of x/y
n root of x * n root of y
n root of x*y
b root of x raise to the a
x^a/b
1^2
1
square root of 1
1
1.4^2
2
square root of 2
approx 1.4
1.7^2
3
square root of 3
approx 1.7
2.25^2
5
square root of 5
approx 2.25
2^2
4
square root of 4
2
3^2
9
square root of 9
3
4^2
16
square root of 16
4
5^2
25
square root of 25
5
6^2
36
square root of 36
6
7^2
49
square root of 49
7
8^2
64
square root of 64
8
9^2
81
square root of 81
9
10^2
100
square root of 100
10
11^2
121
square root of 121
11
12^2
144
square root of 144
12
13^2
169
square root of 169
13
14^2
196
15^2
225
square root of 225
15
16^2
256
square root of 256
16
20^2
400
square root of 400
20
25^2
625
square root of 625
25
30^2
900
square root of 900
30
1^3
1
2^3
8
3^3
27
4^3
64
5^3
125
cube root of 1
1
cube root of 8
2
cube root of 27
3
cube root of 64
4
cube root of 125
5
PEMDAS
Parentheses-Exponents-(mul-div)-(add-sub)
with mul and div & add and sub being performed left to right
;