Number Properties
To estimate the root of an imperfect square
compare the two nearest perfect squares on either side
A number is a perfect square, if its prime factorization
contains only even powers of primes
A number is a perfect cube, if its prime factorization
contains only primes whose powers are multiples of 3
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.
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
2^(1/2) = approximately
1.4
√2≈
1.4
3^(1/2) = approximately
1.7
√3≈
1.7
121^(1/2) =
11
169^(1/2) =
13
196^(1/2) =
14
256^(1/2) =
16
5^(1/2) = approximately
2.25
6^(1/2) = approximately
2.45
7^(1/2) = approximately
2.65
Fewer factors, more multiples.
Any integer only has a limited number of factors. By contrast, there is an infinite number of multiples of an integer
Creating Remainders
Dividend = Quotient * Divsor + Remainder (from X / N = Q + R / N)
Divisible by 6
Divisible by 2 AND 3
The prime factorization of a perfect square contains only ______ powers of primes.
EVEN
The prime factorization of a perfect SQUARE contains only ____ powers of primes.
Even (90^2 = 2^2 3^4 5^2
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 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.
Divisible by 4
Last 2 digits divisible by 4 or 2 twice
Divisible by 2
Last Digit is even
Divisible by 10
Last digit 0
Divisible by 5
Last digit 0 or 5
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.
Divisible by 3
Sum of Digits divisible by 3
LCM
The smallest multiple of two or more integers (Product of all integers in Venn) (Add Fractions)
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)
You can multiply remainders, as long as you
correct excess remainders by dividing the product of the remainders by the divisor, leaving the final remainder
Simple strategy to solve questions about oddness or eveness is
create a table with scenarios as rows, each variable and desired term as columns, fill in boxes with ODD or EVEN
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
In an evenly spaced set, the arithmetic mean (average) and median are
equal to each other, and equal the average of the first and last terms
gcd(m,n)*lcm(m,n) =
gcd(m,n)*lcm(m,n) = mn
An integer has an odd number of total factors iff
it is a perfect square
(The GCF of m and n) x (the LCM of m and n) =
m x n
You can only simplify exponential expressions that are linked by
multiplication or division; You cannot simplify expressions linked by addition or subtraction
Strategy for solving complicated absolute value equations containing two different variables in absolute value expressions is
pick numbers and use a chart with all combos of +/- as rows
Simple strategy to solve questions about positivity or negativity is
pick numbers to test systematically in a table; use each +/- combo as a row, use CRITERION to filter Y/N in column 1, use answer choices as other columns, fill boxes with POS or NEG
The GCF of m and n cannot be larger than
the difference between m and n
The absolute value of a difference (e.g. |x-y|) can be thought of as
the distance between the two numbers
An expression with a negative exponent is
the reciprocal of what that expression would be with a positive exponent.
You can simplify exponential expressions linked by multiplication or division if
they have a base or exponent in common, e.g.:
To find a GCF of two small numbers
use a Venn diagram: find all prime factors of the numbers, place common factors (including copies) into intersection, non-commons (including copies) outside.
To find an LCM of two small numbers
use a Venn diagram: find all prime factors of the numbers, place common factors (including copies) into intersection, non-commons (including copies) outside.
To find the GCF of large numbers, or of more than two numbers
use a chart with prime columns: the NUMBERS as rows, the PRIME FACTORS as columns, fill boxes with the factor to the right exponent (e.g. in the 5 column, 5^3).
To find the LCM of large numbers, or of more than two numbers
use a chart with prime columns: the NUMBERS as rows, the PRIME FACTORS as columns, fill boxes with the factor to the right exponent (e.g. in the 5 column, 5^3).
Simple ways to find all factors of a small number is
use a table of factor pairs (SMALL and LARGE as columns)
Strategy for keeping track of factors of consecutive integers is
use prime boxes (e.g., for x, x+1, x+2)
Simple strategy for using the Factor Foundation Rule is
use prime boxes, or partial prime boxes for constrained unknowns
GCF>?
|m-n|
gcd(m,n) ≥ ______
|m-n|
Prime Numbers: 0x
2,3,5,7
If the problem states/assumes that a number is an integer, check to see if you can use _______.
prime factorization
If estimating a root with a coefficient, _____ .
put the coefficient under the radical to get a better approximation
The product of k consecutive integers is always divisible by
k, and in fact by k factorial (k!)
GCF (m&n) x LCM (m&n)
mXn
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.
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.
Prime Columns
To calculate the GCF we take the SMALLEST count (the lowest power) in any column. To calculate the LCM we take the LARGEST count (the highest power) in any column.
Positive integers with only two factors must be ___.
prime
In an evenly spaced set, the sum of the elements in the set equals
the arithmetic mean / median (average) number in the set times the number of items in the set.
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
In an evenly spaced set, the average can be found by finding ________.
the middle number
The sum of the integers in a set of consecutive integers is a multiple of the number of items iff
the number of items is odd
For ODD ROOTS, the root has ______.
the same sign as the base
The number of terms in an evenly spaced set equals
(Last - First) / Increment + 1
(a^b)x(c^b)
(axc)^b
Prime Numbers: 1x
11,13,17,19
Divisible by 9
Sum of digits divisible by 9
The prime factorization of __________ contains only EVEN powers of primes.
A PERFECT SQUARE
Prime Numbers: 10x
101, 103, 107, 109, 113
The answer choices for data sufficiency problems are
(A) 1 alone is, 2 is not (B) 2 alone is, 1 is not (C) both together are, neither alone is (D) each alone is (E) 1 and 2 together are not
√169=
13
√196=
14
√225=
15
√256=
16
√5≈
2.5
Prime Numbers: 2x
23,29
√625=
25
Prime Numbers: 3x
31,37
ex:// 3ⁿ + 3ⁿ + 3ⁿ = _____ = ______
3·3ⁿ = 3^{n+1}
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
N! is _____ of all integers from 1 to N.
A MULTIPLE
Any integer with an EVEN number of total factors cannot be ______.
A PERFECT SQUARE
Any integer with an ODD number of total factors must be _______.
A PERFECT SQUARE
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
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.
On data sufficiency, ALWAYS _______ algebraic expressions when you can. ESPECIALLY for divisibility.
FACTOR
Consecutive multiples of n Have _____
GCF of n
Prime factors of _____ must come in pairs of three.
PERFECT CUBES
ADD 2 NON-multiples of N
Multiple of N || NON-multiple of N Except when N = 2 (Two ODDs = EVEN)
NI is a ____
Multiple of all the integers from 1 to N.
The prime factorization of a perfect CUBE contains only powers of primes that are ___
Multiples of 3 (90^3 = 2^3 3^6 5^3)
If N is a divisor of x and y, then _______.
N is a divisor of x+y
The two statements in a data sufficiency problem will _______________.
NEVER CONTRADICT ONE ANOTHER
Multiple of N +||- NON-multiple of N
NON-multiple of N. Except when N = 2 (Two ODDs = EVEN)
Divisible by 8
Number formed by last 3 digits is divisible by 8
All perfect squares have a(n) _________ number of total factors.
ODD
The total number of factors of a perfect square is always
ODD
When we take an EVEN ROOT, a radical sign means ________. This is _____ even exponents.
ONLY the nonnegative root of the number UNLIKE
Perfect Squares have an ___ number of total factors
Odd (ie 4 = 4 1 & 2 2) 1 2 4 are factors
Factor (FF)
Positive integer that divides evenly into an LARGER integer (1 2 4 8 are ___ of 8)
Total Factors
Prime Factor then (PowerA + 1) (PowerB + 1) 2000 = 5^3 * 2^4 therefore (3+1) (4+1) = 20
+/- multiple of N
Result is a multiple of N
Quotient
Result of Dividing 2 Integers
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
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.
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 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.
Divisible by 11
The difference between the ODD digits (SUM) - EVEN digits (SUM) is divisible by 11 (includes 0). Ex. 5181. (5+8)-(1+1) divisible by 11
GCF
The largest divisor of two or more integers (Product of Overlap in Venn) (Reduce Fractions)
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.
Integer
Whole Number: -3 -2 -1 0 1 2 3 (Non-Fraction/Non-Decimal)
Factor Foundation Rule: If x is a factor of y, and y is a factor of z, then...
X is a factor z
The formula for finding the number of consecutive multiples in a set is _______.
[(last - first) / increment] + 1
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.
To find the number of factors of a number whose prime factorization is a^b x c^d x
add one to each exponent and multiply the results, as in: (b+1) x (d+1)...
GCF of multiple consecutive mulltiples of n is
n
gcd(k₁n, k₂n) = ______ for integers k₁, k₂
n
Positive integers with more than two factors are ____.
never prime
