Integer Properties Flashcards
1
Q
Factor
A
- a number multiplied to get an integer
- In A * B = C, A and B are factors of C
- 1 is a factor of every integer: 1 * 5 = 5
- Every integer is a factor of itself: 7 * 1 = 7
- Every integer has at least 2 factors
2
Q
Divisor
A
- an integer that divides another integer to get an integer quotient
- In C/A = B, A is a divisor of C
- There is no difference between a factor and a divisor
- 8 is a factor of 24 and a divisor of 24
- Similarly
- 8 is not a factor or divisor of 12, because it results in a fraction, 12/8
3
Q
Finding Factors (Simple)
A
- List the factor pairs
- Factors of 36
- 1 & 36, 2 & 18, 3 & 12, 4 & 9, 6 & 6
- Do this until you run out of pairs
- Factors of 36
4
Q
Negative Factors
A
- Technically divisors and factors could be negative
- -3, -4, -6, -2, -1, -12, are all factors of -12 and 12
5
Q
Divisibility Rule for 2
A
- all even numbers are divisible by 2
- just look at the last digit
6
Q
Divisibility Rule for 5
A
- if the last digit is 5 or 0 the number is divisible by 5
7
Q
Divisibility Rule for 4
A
- if the last two digits of a number, taken as a single number, are divisible by 4 then the entire number is divisible by 4
8
Q
Divisibility Rule for 3
A
- add all individual digits together
- if their sum is divisible by 3 then then the number is divisible by 3
- 234,837 -> 2+3+4 = 9, 8+3+7 = 18 -> 18+9 = 27 (YES!)
9
Q
Divisibility Rule for 6
A
- the number must be…
- divisible by 2 (check last digit)
- divisible by 3 (add digits together)
- 1296 -> 1+2+9+6 = 18 (YES!)
10
Q
Divisibility Rule for 9
A
- add all of the digits together
- if the sum is divisible by 9 then then number is divisible by 9
11
Q
Multiple
A
- a number found by multiplying one of its factors
- 75 and 1250 are multiples of 5
- 63 and 888 are multiples of 3
12
Q
Multiple Idea (General)
A
- If P and Q are multiples of r, then the sum, difference, or product of P and Q are multiples r
- Multiples of 8
- 24 + 80 = 108
- 80 - 24 = 56
- 24 * 80 = 1920
13
Q
Multiple Idea - Multiples of 1
A
- every integer is a multiple of 1
14
Q
Multiple Idea - Integers
A
- to find multiples you multiply the factor times integers
- 3 * {1,2,3,4,5}
15
Q
Multiple Idea - Base Addition & Subtraction
A
- you can simply add or subtract the integer to get the multiple
- 2401 + 7 = 2408
- 2408 + 7 = 2415
- 2401 - 7 = 2393
16
Q
Multiple Idea - Multiple Addition & Subtraction
A
- if P and Q are multiples of r, then (P+Q) and (P-Q) are multiples of r
- 700 + 49 = 749
- 749 +49 = 798
- 700 - 49 = 651
17
Q
Multiple Idea - Multiples of Multiples
A
- if P is a multiple of r, then any multiple of P is a multiple of r
- 52 is a multiple of 13 so any multiple of 52 is a multiple of 13
18
Q
Multiple Idea - Products of Multiples
A
- if P and Q are multiples of r then the product of P*Q is a multiple of r
- 24 and 80 are both multiples of 8, so are the following
- 24 + 80 = 104
- 80 - 24 = 56
- 24*80 = 1,920
19
Q
What is a Prime Number
A
- an integer with only two factors
- prime numbers are the building blocks of all positive numbers
20
Q
Prime Number Facts
A
- 1 is NOT a prime number, since it has only one factor
- 2 is the only even prime number
21
Q
Prime Number Test
A
- test whether a number is divisible by any prime less than 10
- 2, 3, 5, 7
- if not then it is a prime
- for larger numbers find the closest multiples of these numbers then add another to see if it fits
- For 57
- 7*8 + 7= 63 (NO)
- 5*11 + 5 = 60 (NO)
- 3*20 - 3 = 57 (YES)
- For 57
22
Q
Prime Factorization
A
- the prime numbers that multiply to create an integer
- the DNA of a number, revealing all of its ingredients
- an extremely important skill
- Examples
- 9 = 3*3
- 10 = 2*5
- 12 = 3*4 = 3*2*2
- 24 = 8*3 = 2*4*3 = 2*2*2*3 = 23*3
- 100 = 10*10 = 2*5*2*5 = 22*5
23
Q
Prime Factorization (multiples)
A
- if r is a prime factor of Q, then ALL prime factors of r are prime factors of Q
24
Q
Large Number Factoring
A
- find the prime factorization: 8400 = 24 * 3 * 52 * 7
- make a list of the exponents of the prime factors: {4,1,2,3}
- add 1 to every number on the list: {5,2,3,4}
- multiply those numbers together: 5*2*3*4 = 120
25
Finding Odd Number of Factors
* **prime factorize: 21600 = 25 \* 33 \* 52**
* list the exponents of the **odd prime factors bases: {3, 2}**
* **add 1** to each: **{4, 3}**
* multiply them together: **4 \* 3 = 12**
26
Why the procedure for finding the number of factors works
* **21600 = 25 \* 33 \* 52**
* if factor "F" is "built" on these prime factors, how many powers of 2 could F contain? It may contain none (3\*5 = 15). Any from 0 to 5, which is **6 possibilities** {0,1,2,3,4,5}
* Possibilities for base factor 3 {0,1,2,3}
* Possibilities for 5 {0,1,2}
* think of it as **3 slots to build a factor**
* _Slot 1_: factors for 25: 0 - 5 = 6
* _Slot 2_: factors for 33: 0 - 3 = 4
* _Slot 3_: factors for 52: 0 - 2 = 3
27
Squares of Integers
* **perfect squares**
* **exponents** of the prime factors of a square are **all even**
* if **all exponents are even** in a prime factorization then it is a **perfect square**
28
Counting Factors in a Perfect Square
* all exponents are even
* when 1 is added for multiplying all exponents become odd
* so a **perfect square** will always have an **odd number of factors**
29
Multiplying Odd Numbers
* the product of an all odd number problem is an odd number
* 3\*3 = 9
* 5\*7 = 35
* 9\*13 = 117
30
Greatest Common Factor
* also called the Greatest Common Divisor
* the largest factor that two numbers share: 24 & 40 = 8
31
Greatest Common Factor for Large Numbers
* 360 and 800
* **prime factorize:**
* ****360: 23 \* 32 \* 5
* 800: 25 \* 52
* find the biggest factors shared and multiply
* **23 \* 5 = 40**
*
32
Least Common Multiple
* lowest multiple that two numbers share
* 8 & 12 = 24
33
Find the Least Common Multiple
* **prime factorize**: 24 & 32
* 24: 23 \* 3
* 32: 25
* find the GCF: 23
* multiply the GCF \* the **smallest remaining factor**
* 24: 23 \* 3
* 32: 23 \* 4
* multiply each of these factors together: **23 \* 3 \* 4 = 96**
34
Least Common Multiple Symbolic Representation
* M and N are the two numbers and G is the GCF
* N/G = A
* M/G = B
* LCM = A \* B \* G
35
Why learn the Least Common Multiple
* the test will ask for the LCM in **hidden ways**!
* Hot dogs are 8 to a pack and buns are 12 to a pack. What's the least amount of hot dogs to put together with no leftovers? 24
* it is also important for **adding/subtracting fractions**. **the LCM is the** **LCD**
36
Factors and LCM
* if A is a factor of R then the LCM of A and R must be R
* the LCM of 8 and 24 is 24
* if A and B have no factors greater than 1 then the LCM is A\*B
* the LCM of 7 and 15 is 7\*15 = 105
37
GCD LCM Formula
* take **all factors** of one multiple and **multiply** by the **remaining factors of the other**
* 18 and 24
* 18: 32 \* 2 = **3 \* 3 \* 2**
* 24: 23 \* 3 = **2 \* 2** \* 2\* 3 \* 3
* **3 \* 3 \* 2 \* 2 \* 2 = 72**
38
Zero
* **Zero** is neither **positive or negative**
* **Zero** is **even**
39
Odd - Addition Subtraction
* Add/Subtract **unlikes** equals **odd**
* 13 + 8 = 21
* 13 - 8 = 5
40
Even - Addition Subtraction
* Add/Subtract **likes** is **even**
* 5+7 = 12
* 7-5 = 2
41
Odd Even - Even Factor
* If there is an **even factor** in **multiplication** then the number **must be even**
* 3 \* 5 \* 4 = 15 \* 4 = 60
42
Odd Even - Odd Factors
* For a number to be odd **all factors must be odd**
* 3 \* 5 \* 7 = 105
43
Odd Even - Division
* **no hard rules** for **division**
* the **quotient** could be **even, odd,** or a **non-integer**
44
Integer Caution!
* **never assume** a number is an **integer**
* it could be **a fraction** in which case it is **neither odd nor even**
45
Odd Even - Fractions
* **fractions** are neither **odd nor even**