How to find unit digit of powers of numbers:  Pattern 1

If Base's unit's digit is- 2/3/7/ or 8

EX: "Find unit's digit of 233^6"

Bases with Units Digit of 2, 3, 7, or 8 will repeat UDs every 4th value,

Divide the bases's exponent by 4. 

After dividing, 
If remainder is 1, unit digit of number raised to the power 1. 
If remainder is 2, unit digit of number raised to the power 2. 
If remainder is 3, unit digit of number raised to the power 3. 
If remainder is 0, unit digit of number raised to the power 4.

How to find unit digit of powers of numbers (Pattern 2)

Base with Unit's Digit  - 0/1/5/6

Ex: "WHAT'S THE UNIT'S DIGIT OF 256⁷?"

For 0/1/5/ or 6

all subsequent poers of this base will habve the SAME UNIT's DIGIT

How to find unit digit of powers of numbers  Pattern 3

Base with Unit's Digit of 4 

Ex: "What's the Unit's Digit of 724¹⁶?"

For Bases with a Unit's Digit 4....

If power is odd --> unit's digit will be '4' 
If power is even --> unit's digit will be '6'

How to find unit digit of powers of numbers  Pattern 3

For bases with Unit's Digit 9...

If power is odd --> unit's digit will be '9 

If power is even --> unit's digit will be '1' 

How to find unit digit of powers of numbers

What is the unit's digit of 122⁹⁴ ?

Unit's place is 2. So, it repeats every 4th term of the power. 

So, divide the exponent by 4:

94 ÷ 4 = 23, remainder 2 (remainder is what matters). 

Raise the base's unit's digit (2) to the power of 2 (the remainder) to find the unit's digit of 122⁹⁴:

2² = 4, so the Unit's Digit of 122⁹⁴ is 4!

Thus, 4 is the unit's digit of 122^94.

Divisibility Tests: 

How can you determine if N is divisible by 2 ?

unit's digit of n must be 0

(OR)

unit's digit of n must be divisible by 2. 

Divisibility Tests: 

How can you determine if N is divisible by 3 ?

The sum of N's digits must be divisible by 3

Divisibility Tests: 

How can you determine if N is divisible by 4 ?

1.) N's last two digits must read "00" (ex. "1500")

2.) N's last two digits both divisible by 4 (ex."1,284)

Divisibility Tests: 

How can you determine if N is divisible by 6 ?

N must be divisible by both 2 AND 3

Divisibility Tests: 

How can you determine if N is divisible by 8?

1.) Last three digits are zeros (ex. 21,000)

2.) Last three digits of N form a number divisible by 8. 

Finding factors for a POSITIVE INTEGER: 

What is the sum of the positive factors of integer N?

1. Express N in terms of it's prime factors

A^P x B^Q x C^R

2. N will have (P+1)x(Q+1)x(R+1) positive factors

Examples

32= 2⁵. 

Sum of 32's positive factors = (5+1) = 6.

1,452 = 2² x 3 x 11².

Sum of 1452's positive factors = (2+1)x(1+1)x(2+1)

3x2x3= 18 positive factors. 

Find number of factors for a POSITIVE INTEGER: 

Which could be the # of factors for N if N is a perfect square?

a.) 2

b.) 6

c.) 3

c.) 3

Perfect squares will always have ODD # factors

Think about it: N will be "missing" a factor when compared to non-perfect squares because N (and every other perfect square) has a square root that appears twice in its list of factors!! 

Find number of factors for a POSITIVE INTEGER: 

What is the sum of ALL factors of positive nteger N?

1. Express N in terms of it's prime factors

A^P x B^Q x C^R

2.  The sum of N's factors is expressed as: 

[(A^(P+1) - 1)÷(A-1)] x [(B^(Q+1) -1)÷(B-1)] x [(C^(R+1) -1)÷(C-1)]

Factor Properties for Positive Integers

For integer N with prime factorization of a, b, c, d and e, is N a perfect square?

YES

The only numbers with an odd number of prime factors are PERFECT SQUARES!!

REMAINDERS

(I) When 2 numbers (A and B) are divided by same divisor (C) and the remainders obtained are the same, THEN the DIFFERENCE b/w the 2 numbers (A-B)is also divisible by that divisor (A-B)/C

(II) When 2 positive numbers X and Y are divided by the same divisor Z and remainders obtained are 'r1' and 'r2' respectively- - - THEN the remainders obtained when X+Y is divided by Z will be r1+r2 

What is the remainder when 2⁵⁶ is divided by 7 ? 

Using Pattern Method to solve: 

Try to find a pattern by looking at remainders that result when lower values of 2^x are divided by 7:

 1.  2¹/7= remainder 2 (2/7=0. 0+2=2)

2.  2²/7= remainder 4

3.  2³/7= remainder 1

4.  2⁴/7= remainder 2

5.  2⁵/7= remainder 4

Pattern = remainders repeat after 3 exponents.  

56 ÷ 3= 18, remainder 2. The 2^nd term in the pattern will be the answer; therefore: 2⁵⁶ ÷7 has remainder 4!!

Geometry Formulas

Ratios for 45º-45º-90º Triangle

45º-45º-90º

x : x : x√(2)

leg:leg:hypotenuse

Geometry Formulas

Ratios for 30º-60º-90º Triangle

30º-60º-90º

x : x√(3) : 2x

base = x√(3)

Any Equilateral Traingle is comprised of 2 30º-60º-90º Triangles!!

Geometry Formulas

1/2*(LEG)²

Geometry Formulas

Side² x √(3)

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4

Geometry Formulas

1.) Side²

2.)  1/2 * Diagonal²

Geometry Formulas

Base x Height

Geometry Formulas

1) 4 equal length sides

2) Equal Opposite Angles

3) Both sides a parallel

Geometry Formulas

1.  Base x Height

2.  1/2*(Diagonal₁*Diagonal₂)

Geometry Formulas

Rectangle's relationship between Side & Diagonal 

Length²+Width²= Diagonal²

Geometry Formulas

B₁+B₂+S₁+S₂

Geometry Formulas

1/2(Base₁xBase₂)*Height

Geometry Formulas

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Geometry Formulas

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Geometry Formulas

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Geometry Formulas

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