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54 Cards in this Set

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Why do we call it regrouping, or exchanging, and not carrying?
Because we are regrouping 10 ones, for example, into 1 ten (or exhanging 10 ones for 1 ten). We aren't carrying 1 1 over to the tens, as the actual math would indicate.
What are 2 methods for adding in other bases?
Using number cards and expanding the problem.
How do we add in other bases using number cards?
We make cards for the numbers in the units position, base to the power of 1, base to the power of 2, etc. up to the larges place value.

We then put dots on the cards for the corresponding number. The two numbers being added will be one above the other with the place values ligned up.

Add the number of dots from each set onto a new set of place value cards. If the units card has fewer dots than the base, check the next highes place value card

If the units card has the same or greater dots than the base number, regroup the amount of the base number and place one extra dot on the next place value card.

Repeat the lst 2 steps until you have reached the last card on the left of the row.

Translate the dots into a row of numbers. (ie 543 base 6)
How do we subtract in other bases using number cards?
The system is basically the same as for adding. However, if the units number of the minuend is smaller than that of the subtrahend, you have to exchange one dot on the next place card (to the left) for an amount of dots equal to the base unit, on the units card.

Repeat this process as needed until you reach your answer.
What are the three algorithms for multiplication?
Expanded algorithm, instructional algorithm, and final (or compressed) algorithm
How does the expanded algorithm work?
break apart the numbers according to place value of each digit and multiply each digit according to it's place value to obtain partial products which are added to find the final product.
How does the instructional algorithm work?
Multiply each digit according to place value (without separating the numbers first) to obtain partial products which are added to find the final product.
How does the Final Algorithm work?
Mutliply the digits together, and regroup any extra values to the next place value, and add it on. Ex.:

42
x5
___
210 (i carried the 1 from 10 over above the 4..then 5x4+1=21)
What are 2 ways to multiply in another base?
Using place value cards and using the instructional algorithm.
How do we multiply in another base using place value cards.
We make a place value card for units, for base to the power of 1, for base to the power of 2, etc. until we have a card for the highest place value of the first factor.

Just like with addition, you transpose the numbers corresponding to each place value in the appropriate card. Repeat this process for the number equal to the second factor. Ex. If you had 112 base 5 times 3 base 5, you would need three sets of cards with three place value cards each.

Then, just add all the sets of place value cards together, and regroup as necessary.
How do we multiply in another base using the instructional algorithm?
Multiply the numbers together as you would in base 10, but translate the number into the base you are in. Example

42 base 6
x5 base 6
___
010
200
___
210 is wrong because 10 is too large for base 6. it would be 14 base 6. so in base 6 it would be:

42 base 6
x5 base 6
___
014
532
___
546 = 550 base 6
What are the methods we can use for division?
The long division algorithm, the scaffold algorithm, and the short division algorithm (or compact algorithm).
How does the long division algorithm work?
Find out the greatest number of the divisors you can subtract from the dividend (example 723 divided by 5 you would get 100 5's, or 500)

Subtract that number from the total (to get 223, in the example)

Find out the next largest amount of 5's you can subtract (so, following the example, it's 40 5's, for a total of 200), and subtract that from the new total.

Repeat until you can not subtract any more dividends.

Add up the total number of dividends you subtracted (in this case, 144) and whatever the total is that's left over, that is smaller than the dividend, is your remainder. So the answer for 723 divided by 5 is 144 r 3
How does the scaffold algorithm work?
Instead of writing the number of dividends being subtracted to the right of the total dividends (ie. 400 20 5's), it goes above the division symbol.

Add the number of dividends subtracted together to get your total.
How does the compressed algorithm work?
Instead of writing that you are subtracting 400, you write you're subtracting 4. and at the top of the division symbol, instead of writing 20, you write 2.

The numbers all lign up together, to read left to write, the answer.
How do we divide in another base?
Use the scaffold or the long division algorithm, because it is easier to see your work.

You can always verify your answer by converting to base 10.
What is number theory?
the study of the natural numbers and their relationships.
What is the set of natural numbers?
{1,2,3...}

It is denoted with a boldfaced N
What is a factor?
For two whole numbers a,b such that b≠ 0 and a=bc, then b is a factor of a

We can also say that b is a multiple of a, or that b divides a (denoted b|a) or that b is a divisor of a.
What does it mean if b does not divide a?
b does not divide a (denoted b∤a) when b is not a factor of a
What is a prime number?
A natural number that only has 2 factors: 1 and itself.
What is a composite number?
A natural number with more than two factors.
Why is the number one special?
It is neither a prime number nor a composite number.
What is prime factorization
All composite numbers can be written as the product of primes. This is prime factorization.
How many primes are there?
There are infintely many primes.
How do we determine if a number is a prime?
We must determine if it has any factors other than one or itself. To make this simpler, we check if any prime ≤ the square root of the number is a factor.

If none of them are, then the number is a prime. If one or more of them is a factor, then the number is a composite.

This is shown by the theorem "if n is a composite, then there is a prime p≤(square root of n) that p divides n
How many ways can a composite number be prepresented as a product of primes?
One
How do we find how many factors are in a number?
Given the prime factorization of a number, where each prime is in the form of an exponential numberyou mutliply each exponent +1 together.

ex. 2x3x5 has (1+1)(1+1)(1+1) = 2*2*2 = 8 factors
What are the three divisibility properties?
1. If a|b and a|c then a|(b+c)

2. If a|b and a∤c then a∤(b+c)

3. If a|b then a|kb for any whole number k
What else can we say if we know that a|b and b|c?
If a and b have only 1 as a common factor then a|c
What's the test for a 2|n where n is any whole number?
If the last digit of n is 0,2,4,6, or 8 then 2|n
What's the test for 5|n for any whole number n?
If the last digit of n is 0 or 5 then 5|n
What's the test for 10|n for any whole number n?
If 2|n and 5|n then the 10|n.

:. if the last digit of n is 0, 10|n
What's the test for 3|n for any whole number n?
If the sum of n's digits is divisible by 3 then 3|n
What's the test for a 9|n for any whole number n?
If the sum of n's digits is divisible by 9 then 9|n
What's the test for 6|n for any whole number n?
If 2|n and 3|n then 6|n
What's the test for 4|n for any whole number n?
If the last two digits of n are divisible by 4, then 4|n
What's the test for 8|n for any whole number n?
If the last three digits of n are divisible by 8 then 8|n
What's the test for 2 to the power of r divides n for any whole number n?
If the last r digits of n are divisible by 2 to the power of r then 2 to the power of r|n
What's the test for 7|n for any whole number n?
Subtract twice the last digit of n from the new number formed by all the digits of n except the last digit.

Repeat as necessary until you get to a number you reckognize as being divisible by 7.
What's the test for 11|n for any whole number n?
If the sum of the digits in the evens position minus the sum of the digits in the odd position is divisble by 11 then 11|n

**remember: the units position is even**
What is a Greatest Common Divisor?
Given two whole numbers m, n, the greatest natural number d that divides both m and n is the Greatest Common Divisor of m and n

Denoted GCD(m,n) or GCF(M,n) where GCF means Greatest Common Facror
What are three methods for finding the GCD?
Intersection of sets

Prime factorization.

Euclidean Algorithm
How do you find GCD using Intersection of sets?
List the factors for both m and n. Identify the common factors, and select the largest number that is a common factor.
How do you find GCD using Prime Factorization?
Find the prime factorization of both m,n.

Multiply the common factors of both numbers to arrive at the GCD.
How do you find GCD using the Euclidian Algorithm?
Given two positive numbers a,b divide the larger number by the smaller to get a=bq+r

Is the remainder r 0?

If yes, the last divisor is the GCD.

If no, divide the last divisor by the remainder to get a new remainder. Repeat as necessary until the remainder equals 0.
What is a common multiple?
A number is a common multiple of two numbers if it is a multiple of both numbers.
How many common multiples are there for any two natural numbers a and b?
An infinite number
What is the Least Common Multiple?
For any two natural numbers a,b, the least common multiple, denoted LCM(a,b), is the smallest non-zero mutliple of a and b
What are the three ways to find the LCM(a,b)?
Intersection of Sets
Prime factorization
Euclidian Algorithm
How does the Intersection of sets work to find LCM(a,b)?
We list some multiples of each number, starting with 0, identify the common mutliples, and find the smallest one
How does Prime Factorization work to find the LCM(a,b)?
Write down the prime factorization of a and b. LCM(a,b) is built by using each prime factor the maximum number of times it occurs in each number.

Given the prime factorization of both numbers, if a has 2 and b has 2 squared, 2 squared is used to find the LCM(a,b)
How does the Euclidian Algorithm work to find the LCM(a,b)?
Given a,b, use the Euclidian Algorithm to find the GCD(a,b)

The Euclidian Algorithm then says that LCM(a,b)= (ab) divided by (GCD(a,b))
What are "relatively prime" numbers?
When GCD(a,b)=1, a and b are 'relatively prime' because they are both prime numbers.

When GCD(a,b)=1, LCM(a,b)=ab