59 terms

Manhattan GMAT Number Properties

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Prime Numbers:
0x
2,3,5,7
Prime Numbers:
1x
11,13,17,19
Prime Numbers:
2x
23,29
Prime Numbers:
3x
31,37
Prime Numbers:
4x
41,43,47
Prime Numbers:
5x
53,59
Prime Numbers:
6x
61,67
Prime Numbers:
7x
71,73,79
Prime Numbers:
8x
83,89
Prime Numbers:
9x
97
√2≈
1.4
√3≈
1.7
√5≈
2.5
√169=
13
√196=
14
√225=
15
√256=
16
√625=
25
How to test for sufficiency:

If p is an integer, is p/n an integer?
(1) k₁p/n is an integer
(2) k₂p/n is an integer
If gcd(k₁,n) ≠ 1 or gcd(k₂,n) ≠ 1, this proves insufficiency.
On data sufficiency, ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
The two statements in a data sufficiency problem will _______________.
NEVER CONTRADICT ONE ANOTHER
If N is a divisor of x and y, then _______.
N is a divisor of x+y
If the problem states/assumes that a number is an integer, check to see if you can use _______.
prime factorization
The sum of any two primes will be ____, unless ______.
The sum of any two primes will be even, unless one of the two primes is 2.
If 2 cannot be one of the primes in the sum, the sum must be _____.
If 2 cannot be one of the primes in the sum, the sum must be even.
All evenly spaced sets are fully defined if:
1. _____
2. _____
3. _____
are known.
1. The smallest or largest element
2. The increment
3. The number of items in the set
In an evenly spaced set, the ____ and the ____ are equal.
In an evenly spaced set, the average and the median are equal.
In an evenly spaced set, the average can be found by finding ________.
the middle number
In an evenly spaced set, the mean and median are equal to the _____ of _________.
In an evenly spaced set, the mean and median are equal to the average of the first and the last number.
In an evenly spaced set, the sum of the terms is equal to ____.
the average of the set times the number of elements in the set
The formula for finding the number of consecutive multiples in a set is _______.
[(last - first) / increment] + 1
How to find the sum of consecutive integers:
1. Average the first and last to find the mean.
2. Count the number of terms.
3. Multiply the mean by the number of terms.
The average of an ODD number of consecutive integers will ________ be an integer.
The average of an ODD number of consecutive integers will ALWAYS be an integer.
The average of an EVEN number of consecutive integers will ________ be an integer.
The average of an EVEN number of consecutive integers will NEVER be an integer.
The PRODUCT of n consecutive integers is divisible by ____.
The PRODUCT of n consecutive integers is divisible by n!.
The SUM of n consecutive integers is divisible by n if ____, but not if ______.
The SUM of n consecutive integers is divisible by n if n is odd, but not if n is even.
ex:// 3ⁿ + 3ⁿ + 3ⁿ = _____ = ______
3·3ⁿ = 3^{n+1}
When we take an EVEN ROOT, a radical sign means ________. This is _____ even exponents.
ONLY the nonnegative root of the number
UNLIKE
For ODD ROOTS, the root has ______.
the same sign as the base
ex:// ³√216 =
Break the number into prime powers:
216 = 2 * 2 * 2 * 3 * 3 * 3 = 2³ · 3³ = 6³, so
³√216 = ³√6³ = 6
If estimating a root with a coefficient, _____ .
put the coefficient under the radical to get a better approximation
Positive integers with only two factors must be ___.
prime
Positive integers with more than two factors are ____.
never prime
Let N be an integer.
If you add a multiple of N to a non-multiple of N, the result is ________.
a non-multiple of N.
Let N be an integer.
If you add two non-multiples of N, the result could be _______.
either a multiple of N or a non-multiple of N
gcd(m,n)*lcm(m,n) =
gcd(m,n)*lcm(m,n) = mn
gcd(m,n) ≥ ______
|m-n|
gcd(k₁n, k₂n) = ______
for integers k₁, k₂
n
How to solve:

Is the integer z divisible by 6?
(1) gcd(z,12) = 3
(2) gcd(z,15) = 15
Set up prime columns.
-- z 6 12 15
2 --2¹ 2²
3 --3¹ 3¹ 3¹
5 ---------5¹
All perfect squares have a(n) _________ number of total factors.
ODD
Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
Any integer with an EVEN number of total factors cannot be ______.
A PERFECT SQUARE
The prime factorization of a perfect square contains only ______ powers of primes.
EVEN
The prime factorization of __________ contains only EVEN powers of primes.
A PERFECT SQUARE
Prime factors of _____ must come in pairs of three.
PERFECT CUBES
N! is _____ of all integers from 1 to N.
A MULTIPLE
How to solve:

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer n for which 3ⁿ is a factor of p?
(a) 10
(b) 12
(c) 14
(d) 16
(e) 18
Look at the numbers from 1 to 30, inclusive, that have at least one factor of 3 and count up how many each has:
3-1; 6-1; 9-2; 12-1; 15-1; 18-2; 21-1; 24-1; 27-3; 30-1

The answer is C. 14.
How to solve:

For any positive integer n, the sum of the 1st n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301?
(A) 10,100 (B) 20,200 (C) 22,650
(D) 40,200 (E) 45,150
The sum of EVEN INTEGERS between 99 and 301 is
the sum of EVEN INTEGERS between 100 and 300, or
the sum of the 50th EVEN INTEGER through the 150th EVEN INTEGER.

To get this sum:
-Find the sum of the FIRST 150 even integers (ie 2 times the sum of the 1st 150 integers)
-Subtract the sum of the FIRST 49 even integers (ie 2 times the sum of the 1st 49 integers)
How to solve:

If k, m, and t are positive integers and k/6 + m/4 = t/12, do t and 12 have a common factor greater than 1?
1. k is a multiple of 3
2. m is a multiple of 3
Express as 2k + 3m = t.
1. If k is a multiple of 3, then so is t and we have a yes. => S
2. If m is a multiple of 3, we don't know. => I

A/1 Alone.
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