Number Properties Flashcards
What are integers?
Integers are “whole” numbers such as 0,1,2 and 3 that have no fractional part.
Integers can be positive, negative, or zero?
All are correct, integers can be positive (1,2,3…), negative (-1,-2,-3…), or the number 0.
The GMAT uses the term integer to mean a ______ or a ______.
The GMAT uses the integer to mean a NON FRACTION or a NON DECIMAL.
The sum of 2 integers is always an integer. T or F?
TRUE. The sum of 2 integers is ALWAYS an integer.
The difference of two integers is always and integer - True or False?
TRUE. The difference of 2 integers is ALWAYS an integer.
The product of 2 integers is never an integer. T or F?
FALSE. The product of 2 integers is ALWAYS an integer.
The result of dividing two integers is NOT an integer.
FALSE. It’s SOMETIMES an Integer.
Divisible by 2?
- 2, integer is EVEN
Divisible by 3?
- 3, SUM of integers digits is divisible by 3
Divisible by 4?
- 4, divisible by 2 twice OR if last 2 digits are divisible by 4
Divisible by 5?
- 5, integer ends in 0 or 5
Divisible by 6?
- 6, divisible by both 2 AND 3
Divisible by 8?
- 8, divisible by 2 three times OR last 3 digits divisible by 8
Divisible by 9?
- 9, sum of digits divisible by 9
Divisible by 10?
- 10, number ends in 0
Factors and Multiples are essentially WHAT terms?
Opposite
What are factor pairs?
- pairs of factors that yield an integer when multiplied together
How to find a factor pair of X?
- Make a table with 2 columns labeled “small” and “large”
- Start with 1 in the small column and X in the large
- Test the next possible factor of X
- Repeat until the numbers in the small and large columns converge
What is a Factor?
Positive integer that divides evenly into an integer.
- every integer is a factor of itself
- 1 is a factor of every integer
What is a Multiple?
Integer formed by multiplying that integer by any integer.
- negative multiples are possible
- zero is a multiple of every number
- every integer is a multiple of itself
What is the mnemonic to not confuse FACTORS and MULTIPLES?
“Fewer Factors, More Multiples”
Do all of the following say the same thing:
- X is divisible by Y = Y is a divisor of X
- X is a multiple of Y = Y divides X
- X/Y is an integer = X/Y yields a remainder of 0
- X=3(n), n being an integer = Y “goes into” X evenly
Yes!
If you add or subtract Multiples of N, the result is?
a multiple of N.
- N is the divisor of x and y, then N is a divisor of WHAT?
divisor of x + y
What is Prime?
Any positive integer larger than 1 with exactly two factors: 1 and itself.
is 1 Prime?
1 is NOT considered a prime
What is the first prime? And why is it special?
- the first prime is 2, only even prime
first ten primes are…
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
How to find primes?
To find primes, create a prime factor tree.
- factors of N can be found by building all possible products of the prime factors
When to use Prime Factorization?
- Determine if x is divisible by y
- Determine the greatest common factor of two numbers
- Reducing fractions
- Finding the least common multiple of a set of numbers
- simplifying square roots
- Determine the exponent of one side of an equation with integer constraints
What is the Factor Foundation Rule?
If A is a factor of B, and B is a factor of C, then A is a factor of C.
What is a Prime Box?
Holds all prime factors of N.
What can you tell by multiplying factors in a prime box
can tell if X is a factor of Y by multiplying factors in the prime box
What is the Greatest Common Factor (GCF)
GCF: the largest divisor of 2+ integers
Least Common Multiple (LCM)
LCM: the smallest multiple of 2+ integers
If no primes in common, what is the GCF and LCM?
- if no primes are in common, the GCF is 1 and the LCM is the product of the two numbers
On simpler remainder problems, what number is it best to pick?
On simple problems, pick numbers.
- add the desired remainder to a multiple of the divisor
- ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18
Even +/- Even
Even
Even +/- Odd
Odd
Odd +/- Even
Odd
Odd +/- Odd
Even
Even * Even
Even
Even * Odd
Even
Odd * Even
Even
Odd * Odd
Odd
Divisibility of Odds & Evens
NO GUARANTEES!
- can result in odds, evens or non-integers
- odd divided by any number will never be even
- odd divided by even will never equal an integer
What should 3 things should you remember when you see the sum of 2 primes
ALL primes are odd except for 2
- sum of any two primes will always be even unless one of the primes is 2
- if you see a sum of two primes that is odd, one number must be 2
What does absolute value tell you
Tells how far a number is from 0 on the number line and is always positive.
if two numbers are opposite each other, what do they have
if two numbers are opposite each other, they have the same absolute value
Multiplying & Dividing Signed Numbers
If Signs are the Same, the answer is positive but if Not, the answer is Negative.
Evenly Spaced Sets are…
Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16
Consecutive Multiples are…
Special cases of evenly spaced sets: all values in the set are multiples of the increment
- ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
- these sets must be composed of integers
Consecutive Integers are…
Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16
What does absolute value tell you
Tells how far a number is from 0 on the number line and is always positive.
if two numbers are opposite each other, what do they have
if two numbers are opposite each other, they have the same absolute value
Multiplying & Dividing Signed Numbers
If Signs are the Same, the answer is positive but if Not, the answer is Negative.
Evenly Spaced Sets are…
Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16
Consecutive Multiples are…
Special cases of evenly spaced sets: all values in the set are multiples of the increment
- ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
- these sets must be composed of integers
Consecutive Integers are…
Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16
What does absolute value tell you
Tells how far a number is from 0 on the number line and is always positive.
if two numbers are opposite each other, what do they have
if two numbers are opposite each other, they have the same absolute value
Multiplying & Dividing Signed Numbers
If Signs are the Same, the answer is positive but if Not, the answer is Negative.
Evenly Spaced Sets are…
Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16
Consecutive Multiples are…
Special cases of evenly spaced sets: all values in the set are multiples of the increment
- ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
- these sets must be composed of integers
Consecutive Integers are…
Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16
All sets of consecutive integers are sets of
consecutive multiples
All sets of consecutive multiples are
evenly spaced sets
All evenly spaced sets are fully defined if these three parameters are known:
- The smallest (first) or largest (last) number in the set
- The increment (always 1 for consecutive integers
- The number of items in the set
Arithmetic mean
= median.
if there are an even number of elements, the median is
the average of the two middle elements
Mean & median =
average of the first and last element.
Sum of elements =
mean * number of items in the set.
Consecutive integers formula:
(Last - First) + 1 “add one before you’re done”
Consecutive multiples formula:
(Last - First)/Increment + 1
Sum of a set of consecutive integers equals WHAT
The number of itmes in the set times the middle number (aka median of the set)
The product of K consecutive integers is always divisible by
The product of K consecutive integers is always divisible by K factorial (K!)
- ex: 3! = 3x2x1 = 6, always divisible by 3&2
For any set of consecutive integers with an ODD number of items, the sum of all integers is
ALWAYS a multiple of the number of items.
- ex: 1+2+3+4+5=15, multiple of 5
for any set of consecutive integers with an EVEN number of items, the sum of all items is
NEVER a multiple of the number of items.
- ex: 1+2+3+4=10, not a multiple of 4
What can you use to keep track of factors of consecutive integers.
Use prime boxes
What to do with a negative base with exponents
when negative, simply multiply as required
Why beware when there is an EVEN exponent
an EVEN exponent.
- hides the sign of the base - any base raised to an even power is a positive answer
- odd exponents always keep the sign of the
base
base of 0=
=0