Properties of Numbers

Question 1

What are integers?

Which are integers? 3, 5.5, -9, 0, 2.3

Answer 1

Integers are numbers without fractions or decimals.
* Can be positive or negative
* Zero is an integer
* All integers are even (2n) or odd (2n +1). Even defined as being divisible by 2 with no remainder.

3, -9, 0

Question 2

What are whole numbers?

Which are whole numbers? -4, 1, 0, -52, 16

Answer 2

Whole numbers are non-negative integers.
* Thus, positive integers and zero comprise the set of whole numbers.

1, 0, 16

Question 3

Is zero negative or positive?

Is zero even or odd?

Answer 3

Zero is neither negative nor positive.

Zero is even.

Question 4

What are some of the key properties of Zero?

Answer 4

1. Zero, divided by any number other than zero, is zero
2. Anything divided by zero is undefined
3. Zero is the only number that’s neither positive nor negative
4. Zero is an even number
5. Zero is the only number that is equal to its opposite (e.g., 0 = -0)
6. All numbers are factors of zero, and zero is a multiple of all numbers
   (e.g., 3x0 = 0; 0x5 = 0)
7. Zero is not a factor of any number but itself

There are others, but they are common sense (i.e., anything x 0 = 0)

Question 5

What are some of the key properties of One?

Answer 5

• One is a factor of all numbers, and all numbers are multiples of 1
• One is the only number with only 1 factor
• One is not a prime factor

There are others, but they are common sense (i.e., n x 1 = n)

Question 6

How do we define even and odd numbers?

Answer 6

Even: 2n, where n is an integer

Odd: 2n + 1

Question 7

When adding/subtracting two numbers, what’s a trick to determine if the sum/difference is even or odd?

Answer 7

When adding/subtracting two numbers:
* If both are odd or both are even, the sum/difference is even.
* If not, the sum/difference is odd.

Question 8

Sums and Differences That Yield Even and Odd Numbers:

Answer 8

Even
* (even) +/- (even) = (even)
* (odd) +/- (odd) = (even)

Odd
* (even) +/- (odd) = (odd)
* (odd) +/- (even) = (odd)

Question 9

Products That Yield Even and Odd Numbers:

Answer 9

Even
* (even) x (even) = (even)
* (even) x (odd) = (even)

Odd
* (odd) x (odd) = (odd)

Question 10

Even + Odd = ?

Answer 10

Even + Odd = (Odd)

Question 11

Odd + Odd = ?

Answer 11

Odd + Odd = Even

Question 12

Even x Odd = ?

Answer 12

Even x Odd = Even

Question 13

Odd x Odd = ?

Answer 13

Odd x Odd = Odd

Question 14

Odd + Odd = ?

Answer 14

Odd + Odd = Even

Question 15

Quotients That Yield Even and Odd Numbers:

Answer 15

(even) / (even) = (even) or (odd)
(even) / (odd) = (even)

(odd) / (odd) = (odd)

12/6 = 2 ; 12/4 = 3

Question 16

What is a “signed number?”

Answer 16

A signed number refers to a positive or negative number.

Question 17

What is an easy way to add numbers with different signs?

e.g., -9 + 3 = ?

Answer 17

Subtract the absolute value of the smaller number from the larger number, then keep the sign from the larger number.

e.g., -9 + 3 = ?
|9| - |3| = |6| –> -6

Question 18

If we subtract two numbers with opposing signs, how do we approach?
e.g., -7 - 8 = ?

Answer 18

Subtracting is adding the opposite.

We can switch the sign, then use our basic addition rules.

e.g., -7 - 8 = (-7) + (-8) = -15

Question 19

Two rules for even and odd number of exponents (as they related to positive and negative)

Answer 19

Rule 1: When a nonzero base is raised to an even exponent, the result will be even.
e.g., 3^2 = 9 ; (-2)^4 = 16 ; (-1/2)^6 = (1/64)

Rule 2: When a nonzero base is raised to an odd exponent…
if the base is positive, it will be positive.
If the base is negative, the result will be negative.
e.g., 3^3 = 27 ; (-2)^5 = -32 ; (-1/2)^3 = (-1/8)

Question 20

What is a factor/divisor?

Answer 20

A factor (or divisor) is a number that divides evenly into another number.

If y is a factor of x, and x is positive:
* The smallest factor of x will be 1, the largest will be x
* y will fall between 1 and x
* For any positive integers x and y, y is only a factor of x if x/y is an integer.

Question 21

What is a multiple?

Answer 21

A multiple of a number is the product of that number with any integer.
e.g., 2x0 = 0, 2x1 = 2, 2x2 = 4, etc., so 0, 1, 2, 4 are multiples of 2

For two quantities x and y, x is a multiple of y only if x = ny, where n is an integer.

Or: If y=/=0, x is a multiple of y only if x/y is an integer.

Any integer is both a factor and multiple of itself.
e.g., 5 is factor of 5 since 5 divides into 5. 5 is a multiple of 5 since 5x1 = 5

Question 22

What is a prime number?
What is a composite number?

Answer 22

Prime numbers are any numbers greater than 1 with only two factors: 1 and itself.

Composite numbers are non-prime numbers

2 is the only even prime number

Question 23

How do we find a number’s total number of factors?
e.g., 120

Answer 23

Step 1.
Find the prime factorization of the number.
e.g., 120 = 2^3 x 3^1 x 5^1
Step 2.
Add 1 to each of the exponents
e.g., (3+1)(1+1)(1+1)
Step 3.
Multiply these numbers, and you have the number of factors.
e.g., (4)(2)(2) = 16, so 120 has 16 factors

Question 24

If a number is raised to a positive integer exponent, does its number of unique primes increase or decrease?

Answer 24

The number of unique primes does not change.

e.g., 18 = 2^1 x 3^2 … 2 unique primes
Let’s cube 18, giving us 5,832.
5,832 = 2^3 x 3^6 … 2 unique primes

If some number x has y unique prime factors, x^n (where n is a positive integer) will have the same y of unique prime factors.

Question 25

What is the LCM?

e.g., LCM of 2 and 5?
LCM of 12 and 6?

Answer 25

The Least Common Multiple (LCM) of a set of positive integers is the smallest number that is a multiple of each of the integers.

e.g., LCM of 2 and 5 is 10
LCM of 12 and 6 is 48

Question 26

How do we find the LCM of numbers using prime factorization?

e.g., 20 and 45

Answer 26

To find LCM using prime factorization:
1. Put all numbers in prime factor form
2. Of those prime factors repeated, take the one with the larger exponent.
3. Take all integers/prime factors not repeated.
4. Multiply them together to get LCM.

e.g., 20 is 2^2 x 5^1
45 is 3^2 x 5^1

So we take 2^2 x 3^2 x 5^1 = 180

Question 27

When finding the LCM of >2 numbers, how many numbers must a prime factor be shared by to be considered a repeated prime factor?

Answer 27

For a prime factor to be considered a repeated prime factor, it must be shared by at least 2 numbers.

Question 28

If two numbers don’t share any common prime factors, what is their LCM?

Answer 28

If they don’t share prime factors, the LCM is the product of the two numbers.

e.g., 24 and 30 share common factors, so the LCM is 120
6 and 7 share no common factors, so the LCM is 42

Question 29

What is the GCF?

Answer 29

The Greatest Common Factor (GCF) of a set of numbers is the largest number that will divide evenly into all of the numbers.

e.g., GCF of 8, 12, and 16 is 4.

Question 30

How do we find the GCF of a set of numbers?

e.g., GCF of 60 and 72

Answer 30

To find the GCF of a set of numbers:
1. Put all numbers in prime factor form
2. Identify the repeated prime factors
3. Of those repeated, take take only those with the smallest exponents
4. Multiply these to get the GCF
Note: If there are no shared factors, the GCF is 1.

e.g.,
60 = 2^2 x 3^1 x 5^1
72 = 2^3 x 3^2

So we have 2^2 x 3^1 = 4 x 3 = 12 (our GCF)

Question 31

If we know the LCM and GCF of two positive integers (x & y), what else can we find?

e.g., we have positive integers p & q. Their LCM is 240 and their GCF is 8. What is their product?

Answer 31

If LCM (x,y) is p and GCF (x,y) is q, we can determine the product of xy by multiplying them together.

i.e., xy = LCM(x,y) * GCF(x,y)

e.g., xy = LCM(x,y) * GCF(x,y) = 240 * 8 = 1,920

Question 32

Note: if we have the LCM of a set of numbers, we have a list of the unique prime factors of the product in that set.

Question 33

Note: we can use the LCM to solve repeating pattern questions.

e.g., If light L flashes every 32s, light M every 12s, and they flash simultaneous at 8:00 pm, when will they flash together again?

Answer 33

e.g., LCM(light L, light M) = 2^5 x 3^1 = 96s
the amount of time before they flash together again

Question 34

When thinking about divisibility, think about prime factorization.

e.g., if 5^2, 3^2, and 2^2 are factors of 150x, what’s the smallest x can be?

Answer 34

If we prime factor and put 150x in numerator, as well as “factors” in denominator, we see which values we need to satisfy the equation.

Question 35

If x/y is an integer, then any factor of y is also a factor of x.

e.g., 100 / 25 = 4 ; 100 / 5 = 20

Answer 35

That is, if y is a factor of x, and z is a factor of y, z is a factor of x.