Number Props

Question 1

1. Only # = to it’s opposite
2. Multiple of all #’s
3. Not a factor of any # except itself
4. An even number
5. Neither negative nor positive

Answer 1

0 (Properties of Zero)

Question 2

How to find “trailing zeroes” or number of 10s in a large #

Answer 2

Find the number of 2’s and 5’s you have. Whichever number is represented the least amount of times is the number of trailing zeros

Question 3

odd #/odd# =

Answer 3

odd

Question 4

even/odd =

Answer 4

even

Question 5

odd x odd =

or

odd x odd x odd =

Answer 5

odd

Question 6

odd - even =

or

even - odd =

Answer 6

odd

Question 7

odd + even =

or

even + odd =

Answer 7

odd

Question 8

even - even =

Answer 8

even

Question 9

odd - odd =

Answer 9

even

Question 10

odd + odd =

Answer 10

even

Question 11

Odd Integers

Answer 11

Can be represented by (2n+1) or (2n-1)

where “n” is odd

Question 12

Even Integers

Answer 12

Can be represented by (2n) where “n” is an interger Has an even units digit Divisible by 2

Question 13

Prime #’s

Answer 13

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101

Question 14

Total # of factors for a given integer

Answer 14

Step 1: Determine prime factorization for integer

Step 2: Add 1 to the value of each exponent

Step 3: Multiply those values (i.e. “new exponents”), the product is your answer.

Question 15

The # of unique prime factors does not change if…

Answer 15

the number in question is raised to any other positive integer power

Question 16

LCM (Definition)

Answer 16

the smallest positive integer into which all of the numbers in question will divide into

or

the smallest positive multiple of all numbers in the set

Question 17

Finding the LCM of a set of positive integers

Answer 17

Step 1: Prime factorize all integers in set.

Step 2: Of any repeated prime factors, only take the ones with the largest exponent.

Step 3: Take any non repeated prime factors.

Step 4: Multiply numbers from Step 2 & 3. ** If numbers in set do not share any prime factors, then multiply all numbers in the set.

Question 18

Numbers divisible by 4

Answer 18

Last two digits of the number are divisible by 4 Any number that ends in “00” All multiples of 100

Question 19

Numbers divisible by 8

Answer 19

Divide the last 3 digits of even number to determine Any number that ends in 000 All multiples of 1000

Question 20

Numbers divisible by 11

Answer 20

The (sum of the odd-numbered digits) - (sum of the even-numbered digits) = number divisible by 11

Question 21

Numbers divisible by 12

Answer 21

Number is divisible by BOTH 3 & 4

Question 22

Finding the GCF

Answer 22

Step 1: Prime factorize each int. in the set. Put prime factors in exponent form

Step 2: Identify repeated prime factors

Step 3: Of the repeated prime factors, take only the one with the smallest exponent. Discard non-repeated prime factors

Step 4: Multiply numbers found in Step 3 ** If the set has no prime factors in common, the answer is 1.

Question 23

Formula for division w/ a remainder

Answer 23

X = YQ + R

or

Question 24

“4” Units Digit Pattern

Answer 24

4-6-4-6

Question 25

“3” Units Digit Pattern

Answer 25

3-9-7-1

Question 26

“5” & “6”

Units Digit Pattern

Answer 26

All powers of 5 and 6 end in 5 & 6

Question 27

“7” Units Digit Pattern

Answer 27

7-9-3-1

Question 28

“8” Units Digit Pattern

Answer 28

8-4-2-6

Question 29

“9” Units Digit Pattern

Answer 29

9-1-9-1

Question 30

“2” Units Digit Pattern

Answer 30

2-4-8-6

Question 31

If

|x+y| = |x| + |y|

and x & y are both non-zero integers

Answer 31

then…

x & y have the same sign

Question 32

If

|x| - |y| = |x-y|

then…

Answer 32

x & y have the same sign

&

|x| > |y|

Question 33

If x² > b

and b > 0

then….

Answer 33

x > b^1/2

&

x ^1/2

• thus*
• b^1/2 > x > b^1/2

Note: If it’s a > sign the rule works out the same way

Question 34

Determining numerator & denominator from a word problem

Answer 34

look for “is” and “of”

translate as

is/of

or (a/b) x 100

Question 35

(x⁴ - y⁴)

Answer 35

=

(x² - y²)(x² + y²)

Question 36

Notating Even Consecutive Integers

Answer 36

2n, 2n + 2, 2n + 4, 2n + 6, …

Question 37

Notating Odd Consecutive Integers

Answer 37

2n + 1, 2n + 3, 2n + 5, 2n + 7, …

Question 38

(a² - b²)

Answer 38

=

(a+b) (a-b)

Question 39

Square of a Sum

(a+b)²

Answer 39

=

a² + 2ab + b²

Question 40

Square of a Difference

(a-b)²

Answer 40

=

a² - 2ab + b²

Question 41

GCF for Consecutive Integers

n & n+1

Answer 41

No shared factors so the GCF

=

1

Question 42

(LCM of x & y) x (GCF of x & y)

=

Answer 42

Product of x & y

(x)(y)

Question 43

Finding the Unique Primes of Two #’s

(Shortcut)

Answer 43

Find the LCM of both numbers

Question 44

Is this number a perfect square?

Answer 44

If the prime factorization of the number yields

primes^{raised to even expoents}

then it is a perfect square.

Question 45

Terminating vs. Non-terminating Decimals

Answer 45

If the denominator of the fraction in question has a prime fatorization of only 2’s, 5’s or both it will terminate!

Any other primes = non-terminating decimal

Question 46

Division Properties of Factorials

n! must be…

Answer 46

1. Divisible by all integers from 1 to n inclusive
2. Divisible by any combination of integers from 1 to n inclusive

Question 47

For positive integers x & y

Is y a factor of x?

Answer 47

Yes, if x/y is an integer.

Question 48

Product of any 3 consecutive integers

(n)(n+1)(n+2)

or

(n-1)(n)(n+1)

Answer 48

is always divisible by 6

Question 49

Product of any 3 EVEN consecutive integers

(n)(n+2)(n+4)

or

(2n)(2n+2)(2n+4)

or

(n-2)(n)(n+2)

Answer 49

is always divisible by 6

Question 50

The absolute value of the additon of two numbers

|a + b|

will…

Answer 50

Always be less or equal to the absolute value of

a plus the absolute value of b

|a + b| < |a| + |b|

Question 51

The absolute value of the subtraction of two variables

|x - y|

is…

Answer 51

greater than or equal to the subtraction of the

absolute value of the 1st variable minus the absolute value of the 2nd variable

|x - y| > |x| - |y|

Question 52

If the absolute value of an expression is equal to a negative number

|x + 1| = -4

Answer 52

Wait…what?!1/

The absolute value of an expressions can never equal a negative number!

Never

EVER!