Ch. 3 Number Properties Flashcards
What is an integer?
A number that doesn’t have a decimal or fraction:
-100, -20, -1, 0, 3, 13 are all integers
-2.14, Pi, and 1/2 are NOT integers!
Is zero positive or negative?
Neither! There is no sign associated w/ 0
What is a whole number?
Whole numbers are nonnegative integers, so 0 and all positive integers are whole numbers!
What is the only number that is equal to its opposite?
0 = -0
Is zero even or odd?
Even
What is the square root of zero?
0
0^ any positive power = ?
0
0^0 = ?
Undefined, never tested on the GRE
0 * any number = ?
0, so 0 is a multiple of ALL numbers and 0 is equal to all of its multiples
0 is a factor of which numbers?
Only itself, b/c if you multiply anything by 0, it’ll equal 0, so 0 can’t be a factor of 6, for example b/c 0* anything = 0, it can never be 6.
write this algebraically: a is a factor of b
b = a x k, where k is some integer
What number is a factor of ALL numbers?
1
Because k x 1 = k, so 1 is a factor of k
1 ^ any power =
1
if a is any integer, a ^ 0 = ?
1
any number raised to the 0 power is 1 (except 0^0 which is undefined)
ALL numbers are multiples of what number?
1
What numbers have only one factor?
Only 1
All other numbers have at least 1 & itself
Is 1 a prime number?
No, 2 is the first prime number
How can we represent an even integer? An odd integer?
2n for even; 2n +/- 1 for odd
Because all even integers are divisible by 2 w/o a remainder
odd +/- odd =
even
even +/- even
even
even +/- odd
odd +/- even
odd
even x even =
even
even x odd =
even
odd x odd =
odd
If you’re multiplying two or more integers, and any of them are even, the product/ result will be (even/ odd/ can be either one)?
even
What are the results when multiplying evens & odds?
If there are ANY even numbers (no matter how many integers you’re multiplying), then result will be even.
If you’re only multiplying odds, then result is odd.
What are the results when adding/ subtracting evens & odds?
If they’re the same (both even or both odd), then result is even.
If they’re different (even +/- odd), then result is odd.
When an even number is divisible by an even number, is the result even or odd?
It can be either one!
4/2 = 2
6/2 = 3
even / even =
Even OR odd - can be either!
even / odd =
even
odd / odd =
odd
odd / even =
not divisible
What is the absolute value of a number?
The distance between the number & 0 on the number line
When a nonzero base is raised to an even exponent, is the result positive, negative, or either?
Positive, always.
For example,
3^4 = 81
(-2)^2 = 4
(-1/2)^6 = 1/64
When a nonzero base is raised to an odd exponent, is the result positive, negative, or either?
If original base is positive, it’ll be +ve.
If original base is negative, it’ll be -ve.
What does this mean: y is a factor of x?
Y divides evenly into x
2 is a factor of 16
Which number has infinitely many factors?
0
Every number is a factor of 0 b/c every number divides into 0
If a number is positive, what is its smallest factor & largest factor?
1 is smallest, itself is largest, so,
1 <= all factors <= itself
What is another name for factor?
Divisor
2 and 4 are divisors of 16
2 and 4 are factors of 16
For any positive integers x and y, y is a factor of x if and only if what is an integer?
x/y is an integer
Also, 1 <= y <= x
What is a multiple of a number?
A multiple of a number is the product of that number with any integer
Multiple of x = xn
x is a multiple of y only if x = ny where n is an integer
The number x is a multiple of 5, write this algebraically
x = 5n where n is an integer
This means x / 5 is an integer
What is a prime number?
An integer w/ only two factors: one and itself
What are composite numbers?
NOT prime numbers - they have factors OTHER than 1 and themselves
What are the first 25 prime numbers? (1 thru 30)
2, 3, 5, 7
11, 13, 17, 19
23, 29
What are the first 25 prime numbers? (30 thru 60)
31, 37
41, 43, 47
53, 59,
What are the first 25 prime numbers? (60 thru 100)
61, 67,
71, 73, 79
83, 89,
97
How to determine the TOTAL number of factors of a large number? For example: 2,160.
- Find the prime factorization of 2160:
(2^4)(3^3)(5) - Add 1 to the value of each exponent and then multiply the results:
(4+1)(3+1)(1+1) = (5)(4)(2) = 40
2,160 has a total of 40 factors.
If some number x has 3 unique prime factors, how many unique prime factors does x^n have? (where n is a positive integer)
3 unique prime factors
The number of unique prime factors in a number remains constant even when that number is raised to a positive integer power.
For example, 25 has 1 unique prime factor (5)
25x25 still only has 1 unique prime factor (5)
How do you find the LCM of a set of numbers?
- Find the prime factorization of each integer
- If any repeat prime factors among the set, take only the LARGEST exponent
- Take all non-repeat prime factors
- Multiply them all together
Result is LCM
What’s a more painful way to find the LCM?
List out all of the multiples of the largest number in the group; go down the list until you find a number that is divisible by all the numbers in the group.
What is the greatest common factor (GCF)?
The largest number that will divide into a set of numbers.
For example, the GCF of 8, 12, and 16 is 4 because 4 is the greatest number that will divide into those three numbers.
How to find the GCF?
- Prime factorization of each number
- Identify repeat prime factors (if no repeats, then 1)
- Take only the repeats & the smallest exponents
- Multiply together
The result is the GCF
If a set of positive integers has no prime factors in common, what is their GCF?
1
What’s a more painful way to find the GCF?
List all of the factors of one of the numbers; go down the list (from largest to smallest) until you find a factor that the other numbers are divisible by, too.
x and y are any two positive integers, what is
LCM (x,y) x GCF (x,y) = ?
xy
If you have two positive integers, x and y, and you know y divides evenly into x, then what is the LCM and GCF of the two?
If y divides evenly into x (e.g., 3 divides evenly into 6),
LCM (x,y) = x
GCF (x,y) = y
If we know the LCM of a set of positive integers, what else can we find?
All of the unique prime factors of the numbers in the set
Why? Because to determine the LCM, we multiply the largest exponents of the repeat prime factors and all non-repeat prime factors.
So, we can reverse engineer this when we know the LCM
It also provides us with the unique prime factors of the PRODUCT of the numbers in the set
(b/c the prime factors won’t change if we multiply all the numbers)
How can you determine when two processes that occur at different rates or times will coincide?
By using LCM
If you have two positive integers, x and y, what does it mean to say x is divisible by y?
y divides evenly into x
x/y = +ve integer
x/y = z + (no remainder)
x/y = z
In division, what are the different parts called?
x = dividend / numerator
y = divisor / denominator
z is the quotient
What are the different ways to say even division?
- y is a factor of x, y is a divisor of x, y divides into x (evenly)
- 3 is a factor of 6, 3 is a divisor of 6, 3 divides into 6 evenly
- x is a multiple of y, x is a dividend of y, x is divisible by y
- 10 is a multiple of 5, 10 is a dividend of 5, 10 is divisible by 5
Given two positive integers, x and y,
when will x/y yield an integer?
If x is a multiple of y
or if y is a factor of x
When you see a question involving divisibility, what should you immediately think?
PRIME FACTORIZATION!
If x is divisible by y, what else is it divisible by?
All factors of y
If 120 is divisible by 6, then it’s also divisible by factors of 6 (3 and 2)
If x, a, and b are integers, what makes x^a / x^b an integer?
Only if a >= b
If z is divisible by 6 and 4, what is z also divisible by?
12
If z is divisible by x and y, it is also divisible by the LCM(x,y)
That’s because the SMALLEST z could be is 12 (b/c it’s divisible by both 3 and 4, and 12 is divisible by 12)
Divisibility rules: 0, 1, 2, 3, 4, 5, and 6.
- No number is divisible by 0.
- All numbers are divisible by 1.
- All even numbers are divisible by 2.
- Divisible by 3 if sum of all digits is divisible by 3.
- Divisible by 4 if the last two digits are divisible by 4.
- Divisible by 5 if it ends in 0 or 5.
- Divisible by 6 if divisible by both 2 and 3.
Divisibility rules: 8, 9, 10, 11, and 12.
- Divisible by 8 if last three digits are divisible by 8 (including numbers ending in 000)
- Divisible by 9 if sum of all digits is divisible by 9.
- Divisible by 10 if it ends in 0.
- Divisible by 11 if the sum of the odds digits minus the sum of the evens digits are divisible by 11
- Divisible by 12 if divisible by both 3 and 4.
Can you multiply remainders? For example, if we wanted to know the remainder when 12 x 10 is divided by 7
yes! you can divide each number by the divisor, then remove any excess remainder.
For example:
12/7 = 1 + 5/7
10/7 = 1 + 3/7
Multiply 5 x 3 = 15/7
There’s an excess of two 7’s, (15/7 = 2 + 1/7) so your true remainder is 1.
Can you add or subtract remainders?
yes!
Just make sure to account for excess or negative remainders (and keep them in order if you’re subtracting)
When you see a number with units digit of 0, what do you know about its prime factors?
Prime factors include 5 & 2
If 5 & 2 are included in the prime factors, what is the units digit of the number?
0
If you have a number with 4 trailing zeros (k x 10^4), how many (5 x 2) pairs are in that number?
4 pairs in the prime factorization of that number
The product of n consecutive integers must be divisible by what?
n!
(n factorial)
Any factorial >4! has a units digit of what?
- ALLLL factorials greater than 4! MUST have a units digit of 0
5! = 5 x 4 x 3 x 2 x 1 = you have a 5 x 2 pair, so units digit MUST be 0.
All factorials 5! and greater have a 5 x 2 pair so ALL of them have a units digit of 0
A number in between 3 and 7 - is that inclusive of 3 & 7?
No.
Only if it said, “between 3 and 7 inclusive.”
How do you determine the number of digits in a number?
For example, 25^10 x 8^6
- Prime factorize the number
- Count the number of (5 x 2) pairs - each pair contributes ONE trailing zero.
- Collect the number of unpairs 5s or 2s plus any other prime factors and multiply them together. Count the number of digits.
Result is the sum of all of the digits
.002 has how many leading zeros?
2
(number of zeros after decimal point, before first nonzero digit)
When the denominator is NOT a perfect power of 10, and it has 2 digits (e.g., 1/48), how many leading zeros will the decimal have? What about 3 digits? 4 digits?
If den has 2 digits, decimal will have 1 leading zero:
1/80 = 0.0125
1/48 = 0.0208
If den has 3 digits, decimal will have 2 leading zeros:
1/800 = 0.00125
1/400 = 0.0025
If X has k digits, 1/X will have k-1 leading zeros.
X must be an integer and NOT a perfect power of 10
What is a perfect power of 10?
10^ any integer without anything additional
100; 1,000; 10,000 etc.
For example, 12,000 is NOT a perfect power of ten because it’s 10^3 x 12
If X is a perfect power of 10, how many leading zeros will 1/X have?
For example, 1/1000 or 1/(10^3)
1/1000 = 0.001
It has 2 leading zeros
Count the number of ZEROs in the perfect 10, leading zeros will be one less.
Another way to do it is to count the number of digits, k, leading zeros is k-2
10^6 has how many zeros?
6
1,000,000
0! =
1
n! is divisible by what?
all integers from 1 to n inclusive AND any product combos of 1 to n inclusive
The product of any set of n consecutive integers is divisible by what?
For example, product of 3 consecutive integers - 7 x 8 x 9 is divisible by what?
- n factorial, n!
- And (obvs) all integers between 1 and n, inclusive.
- And (obvs) all factors of n!
For example, 7 x 8 x 9 / 3! = 504 / 6
34 x 35 x 36 x 37 x 38 is divisible by 5!
It’s also divisible by 1, 2, 3, 4, and 5, individually
Shortcut to determine the number of x’s (prime) that divide into y!
- Start dividing y by x, x^2, x^3, etc. until the quotient becomes 0
- Add up all the quotients (ignore the remainders)
- Result is the number of prime number x in the prime factorization of y!
For example, you can use this to determine the largest number n in 400! / 5^n
Shortcut to determine the number of x’s (composite or non-prime) that divide into y!
- Break x into prime factors
- Use the largest prime of x & apply the prime shortcut:
a. start dividing y by x, x^2, x^3, etc. until quotient is 0
b. add up all the quotients (ignore remainders)
How do you solve a question like this,
If 100!/(8^n) is an integer, what is the largest possible value of integer n?
8^n = (2^3)^n = 2^3n
100/2 = 50
100/4 = 25
100/8 = 12 + R
100/16 = 6 + R
100/32 = 3 + R
100/64 = 1 + R
100/128 = 0 + R
Add up all quotients: 97
3n <= 97
n <= 32.333
How to count the consecutive numbers from 10 to 99, inclusive?
99 - 10 + 1 = 90
Last number - first number + 1
(b/c you’re including both first & last number)
What are some examples of consecutive integers represented algebraically?
n(n+1)(n-1)
n(n+1)(n+2)
(n+7)(n+8)(n+9)
n^2 + n = n(n+1)
n^3 - n = n(n+1)(n-1)
n^5 - 5n^3 + 4n = n(n+1)(n-1)(n+2)(n-2)
If n is odd, what is (n+1)(n-1)?
both are even, so the product is even
these are two consecutive even integers
The product of n consecutive EVEN integers will always be divisible by what?
2^n x n!
The product of n consecutive ODD integers will always be divisible by what?
nothing - there is no divisibility rule for consecutive odd integers
Perfect squares will NEVER end in which numbers?
2, 3, 7 or 8
A number that ends in 2, 3, 7, or 8 can never be a perfect square.
What is true about the exponents of perfect squares?
Other than 0 and 1, all perfect squares have prime factors with EVEN exponents
For example,
64 = 2^6 –> 6 is even
100 = 5^2 x 2^2 –> 2 is even
What are the first nine non-negative perfect cubes?
0, 1, 8, 27, 64, 125, 216, 343, and 512
What to remember when you see y= x^2 or x^3 and y is an integer?
This is a perfect square or perfect cube; x MUST be an integer
What is true about the exponents of perfect cubes?
Prime factorization of a perfect cube will contain only exponents that are multiples of 3
(except if the perfect cube is 0 or 1)
For example, 64 = 4^3 = 2^6 –> 6 is a multiple of 3
How do you know if a decimal will be terminating or not? For example, will 1/24 terminate?
If the prime factorization of the denominator contains ONLY 2s and/or 5s, it will terminate.
Otherwise, it will not!
What is a terminating decimal?
A decimal that stops, for example, 0.25
It has a finite number of non-zero digits at the end
Non-terminating example = 1/6 = 0.166666666…
How can we determine the remainder when a large number is divided by another number? For example, 3^123 / 4 has what remainder?
You can identify a pattern & determine the remainder that way.
3^1 / 4 = R3
3^2 / 4 = R1
3^3 / 4 = R3
3^4 / 4 = R1
If the exp. is even, R1
If the exp. is odd, R3
Because 3^123 exp is odd, you know the remainder will be 3.
What are some patterns in units digits?
All powers of 0 end in 0
All powers of 1 end in 1
Units digits of positive powers of 2 will follow the pattern: 2-4-8-6
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 2
Units digits of powers of 3 will follow the pattern 3-9-7-1
Units digits of powers of 4 will follow the pattern 4-6. All +ve odd powers end in 4; all +ve even powers end in 6.
What are some patterns in units digits?
All +ve integer powers of 5 -
All +ve integer powers of 6 -
Units digits of +ve powers of 7 -
Units digits of +ve powers of 8 -
Units digits of +ve powers of 9 -
All +ve integer powers of 5 end in 5
All +ve integer powers of 6 end in 6
Units digits of +ve powers of 7 will follow the pattern 7-9-3-1
Units digits of +ve powers of 8 will follow the pattern 8-4-2-6
Units digits of +ve powers of 9 will follow the pattern 9-1
What are the units digit patterns of integers greater than 9?
Same units-digit pattern as the powers of its units digit
For example, 12 will be same as 2 - it will follow units digit pattern of 2-4-8-6.
345 will have the same units digit as 5 - it will end in 5.
If you have this sequence: {-2,-1,0,1,2,-2,-1,0,1,2,…} how can you find the 798th term?
The sequence has a total of 5 numbers, repeating: -2,-1,0,1,2. The 5th term is 2; because there are a total of 5 terms, all terms that are multiples of 5 will be = 2.
10th term = 2
15th term = 2
30th term = 2
200th term = 2
800th term = 2
From here, you can count backwards using the sequence:
799th term = 1
798th term = 0
When a whole number is divided by 10, what is the remainder?
The remainder will be the UNITS digit of the numerator.
So, 217/10 –> R7 b/c 7 is units digit of 217
(255x255)/10 –> R5 b/c 255 x 255 ends in ..25, and that units digit is 5.
When a whole number is divided by 1000, what is the remainder?
The last three digits of the dividend.
4102/1000 –> R102
What’s an easy way to calculate the remainder here? and why? 41,503/5
When integers with the same units digit are divided by 5, the remainder is the same.
So, for example, 17/5, 117/5, and 3207/5 will all have the same remainder (which is 2)
In the same vain, 3/5, 13/5, and 41,503/5 will all have the same remainder, which is 3.
What is an evenly spaced set?
The numbers in the set increase by the same amount and share a common difference:
{11,22,33,44,…}
Each term increases by 11; 11 is their common difference.
What are the three ways that evenly spaced sets typically show up on the GRE?
- Consecutive integers (including even & odd consecutive integers)
- Consecutive multiples of a given number such as {2,4,6,8} or {33,66,99,132}
- A set of consecutive numbers with a remainder when divided by some integer such as {1,6,11,16,21} or {3,7,11,15,19}
What is the GCF of two consecutive integers? Why?
1
That’s b/c two consecutive integers NEVER share the same prime factors!
