Number Properties, Prime Numbers, Prime Factorization

Understanding decimal places

.0234 = .234/10 = 2.34/100 = 23.4/1000 = 234/10,000

First 9 non-negative perfect cubs

0, 1, 8 = 2, 27 = 3, 64 = 4, 125 = 5, 216 = 6, 343 = 7, 512 = 8

All perfect squares must end in

0, 1, 4, 5, 6, 9

First 9 perfect squares

0, 1, 4, 9, 16, 25, 36, 49, 64

Divisibility Rules of 2

2 = any number that ends in 0, 2, 4, 6 or 8 is divisible by 2 ex: 456, 791, 824 - not 34,807

Examples of even division factors and multiples

2 divides into 4, 2 is a factor of 4, 4 is a multiple of 2, 5 divides into 100, 5 is a factor of 100, 100 is a multiple of 5, 8 divides into 24, 8 is a factor of 24, 24 is a multiple of 8

Prime numbers less than 100

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

Power pattern of 2

2, 4, 8, 6

Power pattern of 3

3, 9, 7, 1

Power pattern of 4

4, 6

Power pattern of 5

5

Power pattern of 6

6

Power pattern of 7

7, 9, 3, 1

Divisibility Rules of 11

Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11. 1364 (+1−3+6−4 = 0) Yes 913 (+9−1+3 = 11) Yes 3729 (+3−7+2−9 = −11) Yes 987 (+9−8+7 = 8) No

Any Factorials greater than or equal to 5

Any factorial greater than or equal to 5 will have zero as it's units digit

M=2n + 1

Definition of odd number

LCM x GCF

If the LCM of x and y is p and the GCF of x and y is q, then xy = pq

Divisibility Rules of 12

If the number is divisible by both 3 and 4, it is also divisible by 12.

Leading Zeros in a Decimal

If x is an integer with k digits, then 1/x will have k-1 leading zeros unless x is a perfect power of 10, in which case there will be k-2 leading zeros

Factors

If y divides evenly into x, we say y is a factor of x ex: what are the factors of 16? 1, 2, 4, 8, and 16

Divisibility Rules of 6

Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above) 114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes 308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No

Divide by 5 Rule

Step 1: Double Number Step 2: Divide by 10 ex: 630/5 = 630x2 = 1260/10 = 126

Finding the LCM

Step 1: Find the prime factorization of each integer. That is, prime factorize each integer and put the prime factors of each integer in exponent form Step 2: Of any repeated prime factors among the integers in the set, take only those with the largest exponent. If we had 3^2 and 3^3, we'd choose 3^3. If we're left with two of the same power, just take that number once Step 3: Of what is left, take all non-repeated prime factors of the integers Step 4: Multiple together what you found in steps 2 and 3. The result is the least common multiple. ex: LCM of 24 and 60 Step 1: 24 = 2^3 x 3^1 60 = 2^2 x 3^1 x 5^1 Step 2: 2^3, 3^1 Step 3: 5^1 Step 4: 8x3x5 = 120

Doubling and Halving Rule

Step 1: when a number ends in 5 or 50 and the other number is even Step 2: divide the even number by 2 and multiple the odd number by 2 ex: 16x35 = (8x2) x35 = 8x70 = 560

Divisibility Rules of 9

The sum of the digits is divisible by 9 (Note: This rule can be repeated when needed) 1629 (1+6+2+9=18, and again, 1+8=9) Yes 2013 (2+0+1+3=6) No

Two Consecutive Integers

Two consecutive integers will never share any prime factors. Thus, GCF of two consecutive integers is 1.

If you divide or multiply both sides of an inequality by a negative number

You must reverse the direction of the inequality sign. Ex: -x > 4, when dividing by -1 = x < -4

Multiplication Rules for Even and Odd Numbers

even x even = even even x odd = even odd x even = even odd x odd = odd

Division Rules for Even and Odd Numbers

even/odd = even odd/odd = odd even/even = even or odd

Division rules for even and odd numbers when numbers divide evenly into each other

even/odd is even, odd/odd is odd, odd/even =/= because odd numbers cannot be divisible by even numbers

Resulting signs when multiplying or dividing by positive and negative values

if x is positive and y is positive, xy or x/y is positive, if x is negative and y is negative, xy or x/y is positive, if x is positive and y is negative, xy or x/y is negative, if x is negative and y is positive, xy or x/y is negative

Even/Odd Rules for Addition/Subtraction

odd + odd = even even + even = even even + odd = odd odd - odd = even even - even = even even - odd = odd odd - even = odd

Multiplication and Division of Numbers with Signs

positive x positive = positive negative x negative = positive positive/positive = positive negative/negative = positive positive x negative = negative positive/negative = negative

A formula for division

x/y = quotient + remainder/y ex: 23/5 = 4+3/5 solve for x: Q y + r solve for Q: x-r/y solve for r: x-Qy

hints to know that dividing some number x by some number y will result in an integer

y is a factor of x, y is a divisor of x, y divides into x (evenly), x is a multiple of y, x is a dividend of y, x is divisible by x

Multiples

A multiple of a number is the product of that number and any integer ex: what are the multiples of 4? 4, 8, 12, 16, 20...4n

The Range of Possible Remainders

A reminder must be a non-negative integer that is less than the divisor

Power pattern of 8

8, 4, 2, 6

Power pattern of 9

9, 1

Divisibility Rules of 10

The number ends in 0 220 Yes 221 No

Trailing Zeros

The number of trailing zeros of a number is the number of (5x2) pairs in the prime factorization of that number ex: 520 can be expressed as 52x10 = 52 x (5x2) and thus has one trailing zero ex: 5,200 can be expressed as 52 x 100 = 52 x 10^2 = 52 x (5x2)^2 and has two trailing zeros

Finding the GCF

Step 1: Find the prime factorization of each number. That is, prime factorize each number and put the prime factors of each number in exponent form Step 2: Of any repeat prime factors among the numbers, take only those with the smallest exponent. (If no repeated prime factors are found, the GCF is 1.) Step 3: Multiply together the numbers that you found in step 3 The product is the GCF. ex: GCF of 24 and 60 Step 1: 24 = 2^3 x 3^1 60 = 2^2 x 3^1 x 5^1 Steps 2 and 3: 2^2, 3^1 Step 4: 4x3 = 12

Finding the Number of Factors in a Particular number

Step 1: Find the prime factorization of the number Step 2: Add 1 to the value of each exponent. Then multiply these results and the product will be the total number of factors for that number ex: the number of factors of 240 240 = 2^4 x 3^1 x 5^1 = (4+1) x (1+1) x (1+1) = 240 has a total of 20 factors

Divisibility Rules of 7

Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7. 672 (Double 2 is 4, 67-4=63, and 63÷7=9) Yes 105 (Double 5 is 10, 10-10=0, and 0 is divisible by 7) Yes 905 (Double 5 is 10, 90-10=80, and 80÷7=11 3/7) No

Terminating Decimals

The decimal equivalent of a fraction will terminate if and only if the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s, or both ex: 1/20 = 0.05 ex: 1/12 = 0.08333333....

Divisibility Rules of 4

The last 2 digits are divisible by 4 1312 is (12÷4=3) Yes 7019 is not (19÷4=4 3/4) No

Divisibility Rules of 5

The last digit is 0 or 5 175 Yes 809 No

Divisibility Rules of 8

The last three digits are divisible by 8 & the numbers ends in three 000s 109816 (816÷8=102) Yes 216302 (302÷8=37 3/4) No A quick check is to halve three times and the result is still a whole number: 816/2 = 408, 408/2 = 204, 204/2 = 102 Yes 302/2 = 151, 151/2 = 75.5 No

Divisibility Rules of 3

The sum of the digits is divisible by 3 381 (3+8+1=12, and 12÷3 = 4) Yes 217 (2+1+7=10, and 10÷3 = 3 1/3) No