TTP - Math - Linear and Quadratic Equation

Numbers divisible by 11 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 12 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 2 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 3 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 4 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 5 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 6= _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 8 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 9 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 0 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4.

For xa/xb, if a≥b then result is an ______. If b≥a, then result is a _____.

Divisibility with exponents For xa/xb, if a≥b then result is an integer. If b≥a, then result is a fraction.

5x + 8y = 55, what is x

Equation traps When one equation can determine value for 2 variables To find x, group common multiples on one side of equation, then determine which of the numerators are divisible by denominator 5x + 8y = 55, what is x 8y = 55 - 5x y = 5(11-x)/8, 11-x has to be divisible by 8 x = 3 y = 5

E + E = ____ O + O = ____ E - E = ____ O - O = ____

Even/odd numbers Rule for addition, subtraction, multiplication, division E + E = E O + O = E E - E = E O - O = E E + O = O O + E = O E - O = O O - E = O

E + O = ____ O + E = ____ E - O = ____ O - E = ____

Even/odd numbers Rule for addition, subtraction, multiplication, division E + E = E O + O = E E - E = E O - O = E E + O = O O + E = O E - O = O O - E = O

E / E = ____ O / O = ____ E / O = ____ O / E = ____

Even/odd numbers Rule for addition, subtraction, multiplication, division E x E = E E x O = E O x O = O E / E = E or O O / O = O E / O = E O / E = not an integer

E x E = ____ E x O = ____ O x O = ____

Even/odd numbers Rule for addition, subtraction, multiplication, division E x E = E E x O = E O x O = O E / E = E or O O / O = O E / O = E O / E = undefined

34 x 35 x 36 x 37 x 38 / ___! = integer

Factorial notation The product of any set of consecutive inters is divisible by any of their integers in the set 4! = 4 x 3 x 2 x 1, 4! Divisible by 4 x 2, or 3 x 1 The product of any set of n consecutive positive integers is divisible by n!. 34 x 35 x 36 x 37 x 38 / 5! = integer 31 x 32 x 33 x 34 x 35 / 5! = integer

If x/y = positive integer, then any factor of ___ is a factor of ___

Factors of factors If x/y = positive integer, then any factor of y is a factor of x

Find the Greatest Common Factor (GCF)

Greatest common factor (GCF) Is the largest number that will divide into all numbers of the set GCF of 8, 12, 14 = 2 Finding GCF (Method 1) Step 1: Find the prime factorization of each integer Step 2: Identify repeating prime factors Step 3: Find repeating prime factors with the smallest exponent Step 4: Multiple these numbers

Zero product property

If 2 quantities multiply to 0, then at least one of the properties is equal to 0 If (a-b)(c-d) = 0, then (a-b) = 0 or (c-d) = 0

If z is divisible by both x and y, z must also be divisible by _____ of x and y

If z is divisible by both x and y, z must also be divisible by LCM of x and y

Important facts of LCM and GCF: Fact 1 of 4

Important facts of LCM and GCF Fact 1: The LCM and GCF when one number divides evenly into the other number If positive integer y divides evenly into positive integer x, then LCM is x, and GCF is y

Important facts of LCM and GCF: Fact 2 of 4

Important facts of LCM and GCF Fact 2: If we know LCM and GCF of 2 positive integers, x and y, we know product of x and y If LCM of x and y is p and GCF of x and y is q, then xy = pq

Important facts of LCM and GCF: Fact 3 of 4

Important facts of LCM and GCF Fact 3: The LCM provide us with all the unique prime factors of some set of positive integers The LCM of a set of positive integers provides us with all the unique prime factors of the set. Thus, it also provides all the unique prime factors of the product of the numbers of the set For example, for integers x, y, and z, LCM = 360. Prime factorization of 360 is 2*3, 3*2, 5*1. Thus any factor of 360 contains only these 3 prime factors.

Important facts of LCM and GCF: Fact 4 of 4

Important facts of LCM and GCF Fact 4: LCM can be used to solve repeating pattern questions The LCM can be used to determine 2 processes that occur at differing rates or times will coincide Example: Light 1 flashes every 32 seconds, Light 2 flashes every 12 seconds. The time the lights flash together is LCM of two numbers, which is at 96 seconds

Prime numbers to 50

Memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 57, 59, 61, 67, 71, 73, 79, 83, 87, 91, 97

Prime numbers to 50 to 100

Memorize: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 57, 59, 61, 67, 71, 73, 79, 83, 87, 91, 97

What is remainder of 12x13x17/5 =

Multiplying remainders 12x13x17/5 = 12/5 x 13/5 x 17/5 = remainder 12/5 x remainder 13/5 x remainder 17/5 / 5 = product of remainders / 5 = remainder

Name 2 types of number patterns for solving large number problems. Describe a method to determine these patterns

Number patterns 1. Remainders exhibit patterns by divisors 2. Remainders exhibit patterns by powers Solution: Do test cases of small numbers to look for pattern, then apply those pattern to large, difficult to calculate numbers

What is the units digit for 1*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 1*455 = units digit is 1

What is the units digit for 2*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 2*455 = 455/4 = 113 3/4 = units digit is 8

What is the units digit for 4*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 3*455 = 455/2 = odd = units digit is 4

What is the units digit for 3*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 3*455 = 455/4 = 113 3/4 = units digit is 8

What is the units digit for 5*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 5*455 = units digit is 5

What is the units digit for 6*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = 4-6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 6*455 = units digit is 6

What is the units digit for 7*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = Positive odd powers end in 6. Even powers end in 6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 7*455 = 455/4 = 113 3/4 = units digit is 3

What is the units digit for 8*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = Positive odd powers end in 6. Even powers end in 6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 8*455 = 455/4 = 113 3/4 = units digit is 2

What is the units digit for 9*455

Patterns of units digits for consecutive exponents for positive integers, number*exponent Number 0 = 0's Number 1 = 1s Number 2 = 2-4-8-6 Number 3 = 3-9-7-1 Number 4 = Positive odd powers end in 6. Even powers end in 6 Number 5 = 5 Number 6 = 6 Number 7 = 7-9-3-1 Number 8 = 8-4-2-6 Number 9 = Positive odd powers end in 9. Even powers end in 1 9*455 = 455/2 = odd = 9

The prime factorization of perfect cubes has exponents that are multiples of ___

Perfect cubes Prime factorization of perfect cubes, other than 0 and 1, have exponents that are multiples of 3. Prime factors of 450p has to have prime exponents in multiples of 3. Prime factorization of 450 = 2*1x3*2x5*2 Minimum prime factorization of p = 2*2x3*1x5*1 = 60

Perfect squares end in: ___, ___, ___, ___, and ___.

Perfect squares Perfect square: 4, 9, 16, 25, 36, 49, 64, 81 Perfect squares end in: 1, 4, 5, 6, and 9. Perfect squares never end in 2, 3, 7, or 8 Prime factorization of perfect squares, other than 0 and 1, have only even exponents. 64 = 2*6 25 = 5*2

Perfect squares never end in ___, ___, ___, or ___

Perfect squares Perfect square: 4, 9, 16, 25, 36, 49, 64, 81 Perfect squares end in: 1, 4, 5, 6, and 9. Perfect squares never end in 2, 3, 7, or 8 Prime factorization of perfect squares, other than 0 and 1, have only even exponents. 64 = 2*6 25 = 5*2

Prime factorization of perfect squares, other than 0 and 1, have only _____ exponents.

Perfect squares Perfect square: 4, 9, 16, 25, 36, 49, 64, 81 Perfect squares end in: 1, 4, 5, 6, and 9. Perfect squares never end in 2, 3, 7, or 8 Prime factorization of perfect squares, other than 0 and 1, have only even exponents. 64 = 2*6 25 = 5*2

If p and q are positive integers, 450p = q*3. What are the minimum prime factorization for p?

Prime factors of 450p have to have prime exponents in multiples of 3. Prime factorization of 450 = 2*1x3*2x5*2 Minimum prime factorization of p = 2*2x3*1x5*1 = 60

What is range of possible remainders

Range of possible remainders Divisor - 1 The remainder is a non-negative integer less than divisor

What is the rule for remainders of division by 10*n

Remainders after division by 10*n When a whole number is divided by 10, the remainder is the units digit. When the number is divided by 100, the remainder is the last two digits. When the number is divided by 1000, the remainder is the last 3 digits Example 153/10 = 15 3/10 153/100 = 1 53/100 153/1000 = 0 153/1000

What is the rule for remainders of division by 5

Remainders after division by 5 When integers with the same units digit are divided by 5, the remainder remains constant Example 9/5 = 1 ⅘ 19/5 = 3 ⅘ 29/5 = 5 ⅘

Q = x - r/y

Reminder theory Formula for division x/y = Q + r/y → x = Qy + r r = x - Qy Q = x - r/y

r = x - Qy

Reminder theory Formula for division x/y = Q + r/y → x = Qy + r r = x - Qy Q = x - r/y

x = Qy + r

Reminder theory Formula for division x/y = Q + r/y → x = Qy + r r = x - Qy Q = x - r/y

x/y = Q + r/y

Reminder theory Formula for division x/y = Q + r/y → x = Qy + r r = x - Qy Q = x - r/y

What is number of primes in a factorial when base number is a power of a prime number. Solve for number of primes in factorial 30!/4!

Shortcut for determining number of primes in a factorial when the base of the divisor is a power of a prime number 30!/4! = 30!/(2*2)*n = 30!/4*2n Step 1: Determine number of factors of 2 in 30! 30/2, 30/4, 30/8, 30/16, etc = 15, 7, 3, 1 = 26 = 2*26 Step 2: Create inequality of n! with 2n 2n ≤ 26 n ≤ 13 40!/2s = 20 + 10 + 5 + 2 = 38

What is number of primes in a factorial when base number is not a prime number. Solve for number of primes in factorial 40!/6!

Shortcut for determining number of primes in a factorial when the base of the divisor is not a prime number 40!/6! Step 1: Break 6! Into prime factors pairs, 2 x 3 prime factor pair of 6 Step 2: Determine number of (2 x 3) factor pairs in 40!. Start with 2s 40!/2s = 20 + 10 + 5 + 2 + 1 = 40 Then do same with 3s 40!/3s = 13 + 4 + 1 = 18 Number of possible factor pairs of (2 x 3) is 18

What is the number of primes in the factorial 21!/3n?

Shortcut for determining the number of primes in a factorial x!/y Step 1: Divide y by x1, x2, x3 , etc. Keep track of quotients and ignore remainders. Step when the quotient is zero Step 2: Add quotients together. The sum will be a number of x prime factors in y! Example 21!/3n Divide 21 by all 3*1, 3*2, 3*3, 3*4 until quotient is 0 Take all those quotients and add together: 21/3*1, 21/3*2 --> 7, 2, = 9 There are 9 three primes in 21!

Finding the Least Common Multiple (LCM)

Smallest positive multiple of all integers in set. Finding LCM (Method 1) Step 1: Find the prime factorization of each integer Step 2: Find all common prime factors with the largest exponents. Step 3: Find all non-repeated prime factors Step 4: Multiply repeating prime with non-repeating prime factors

29Q/17 = integer 29 is not divisible by _____, then Q must be divisible by _____

Splitting up x to determine if y is a factor 29Q/17 = integer 29 is not divisible by 17, then Q must be divisible by 17

Finding the number of factors in a particular number

Step 1: find prime factorization Step 2: Add "1" to each prime factor exponent Step 3: Multiple exponents

Is 1/12 a terminating decimal?

The decimal of a fraction will terminate if the denominator has a prime factorization containing only 2s and 5s or both. Example 1/25, 25 has prime factors of 5*2 → terminating decimal (0.04) Example 1/32, 32 has prime factors of 2*5 → terminating decimal (0.03125) Example 1/12, 12 has prime factors 2*2 x 3*1 → non terminating decimal (0.08333333)

Is 1/32 a terminating decimal?

The decimal of a fraction will terminate if the denominator has a prime factorization containing only 2s and 5s or both. Example 1/25, 25 has prime factors of 5*2 → terminating decimal (0.04) Example 1/32, 32 has prime factors of 2*5 → terminating decimal (0.03125) Example 1/12, 12 has prime factors 2*2 x 3*1 → non terminating decimal (0.08333333)

What is the rule regarding consecutive integers

Two consecutive integers will never share the same prime factors GCF of two consecutive integers is 1 GCF(n, n + 1) = 1

Quadratic equations that are hidden within fractions

Use LCM to get rid of fraction and be left with quadratic equation to solve for x

How many digits in 50*8 x 8*3 x 11*2?

Using trailing zeros to determine number of digits in an integer Convert trailing zeros into 5 x 2 pairs and then add remaining number Step 1: Prime factorize number Step 2: Count number of 5 x 2 pairs to give trailing zeros Step 3: Multiple remaining prime factors to give number of pre zero digits Step 4: Sum pre zero digits plus trailing zero PF = 50*8 --> 5*8 x 2*8 x 5*8 --> 2*8 x 5*16 PF = 8*3 --> 2*9 PF = 11*2 PF = 2*17 x 5*16 x 11*2 --> 10*16 x 2 x 121 --> 16 + 3 digits = 19 digits

Two rules related to LCM

When finding LCM (of 3 or more numbers), repeated prime factors need not be shared by ALL numbers. Only need to be shared by two numbers. If 2 positive integers, x and y, share no prime factors, the LCM of x and y is xy. 24 and 30 = 120 6 and 7 = 42

If item went on sale for 16% of original N = 16/100 W, W = ___N/___. N must be a multiple of ___

Word problems involving divisibility If item went on sale for 16% of original N = 16/100 W, W = 25N/4. N must be a multiple of 4

Differences of square

x*2 - 1 = (x-1)(x+1) x*2 - 9 = (x-3)(x+3) 3*30 - 2*30 = (3*15 - 2*15)(3*15 + 2*15)

3 common quadratic equations

(x+y)(x+y) = x*2 + 2xy + y*2 (x-y)(x-y) = x*2 - 2xy + y*2 (x+y)(x-y) = x*2 - y*2

An equation has three or more solutions

1. Factoring out the greatest common factor (GCF) - Pulling out x exponents to create a quadratic equation 2. Factoring by grouping - Rearrange x exponents, pull out exponents to get a quadratic equation 3. Assuming the value of a variable cannot be zero x*2 = 100x does not equal x = 100 x*2 = 100x x*2 - 100x = 0 x(x-100) = 0, thus x = 0 or 100

Prime factorization

Any number broken down into prime number factors

How many decimal leading zeros in 1/5*5 x 2*11

Determining the number of leading zeros in decimals If x is an integer with k digits If x is not a perfect power of 10 1/x will have (k-1) leading zeros in its decimal form. If x is a perfect power of 10 1/x will have (k-2) leading zeros in its decimal form. 5*5 x 2*11 = 10*5 x 2*6 = 100,000 x 32 = 3,200,000 --> 7 digits --> k-1 --> 7 - 1 --> 6 leading zeros

How many decimal leading zeros in 1/5*5 x 2*5

Determining the number of leading zeros in decimals If x is an integer with k digits If x is not a perfect power of 10 1/x will have (k-1) leading zeros in its decimal form. If x is a perfect power of 10 1/x will have (k-2) leading zeros in its decimal form. 5*5 x 2*5 = 10*5 x 2*5 = 100,000 --> 6 digits --> k-2 --> 6 - 2 --> 4 leading zeros

How to determine the trailing zeros - Fact 1

Determining the trailing zeros Fact 1: Trailing zeros are created by 5 x 2 pairs Fact 2: Any factorial ≳ 5! will always have a 0 as its unit digit

How to determine the trailing zeros - Fact 2

Determining the trailing zeros Fact 1: Trailing zeros are created by 5 x 2 pairs Fact 2: Any factorial ≳ 5! will always have a 0 as its unit digit

Numbers divisible by 1 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4

Numbers divisible by 10 = _____

Divisibility rules Numbers divisible by 0 = none Numbers divisible by 1 = all Numbers divisible by 2 = if evens Numbers divisible by 3 = if sum of digits is divisible by 3 Numbers divisible by 4 = if last 2 digits divisible by 4 Numbers divisible by 5 = last digit 5 or 0 Numbes divisible by 6 = combine rules for 2 and 3. Add all digits and see if even Numbers divisible by 7 = no rule Numbers divisible by 8 = last 3 digits are divisible by 8 Numbers divisible by 9 = sum of digits is divisible by 9 Numbers divisible by 10 = end in 0 Numbers divisible by 11 = odd-numbered place digits minus sum of the even-numbered place digitis divided by 11. Numbers divisible by 12 = Combine rules for 3 and 4