The GCF and the LCM

To me, I have always had problems with struggling to find the GCF and the LCM. It seemed very easy to me at first but when we started to find the LCM when i was in middle school I struggled all the way up to college because I couldn't understand the the way my teachers taught me. But know after spending some time on it, I have learned an easier ways to find the GCF and the LCM in my math class. To start out we need to know what a prime factorization is. Prime Factorization is finding which prime numbers you need to multiply together to get the original number. A great website to look at examples is found here. Another easy way to find the prime factorization is to use a factor tree which takes the number and takes any two factors and then takes their factors leaving you at the bottom when you can factor no more the prime numbers, which when multipled together give you the original number. A great website to try different numbers using a factor tree is found here.

So GCF stands for Greatest Common Factor, and LCM stands for the Least Common Multiple. The Greatest Common Factor can be found between two numbers and it is the greatest whole number that divides evenly into each of the numbers.
Here is a video that helps you find the GCF.
Here is a video that helps you find the LCM.
To find the LCM-the least common multiple of two or more non-zero whole numbers is actually the smallest whole number that is divisible by each of the numbers. Listed here is a website that teaches you how to find the Least Common Multiple. Here is an example of finding the LCM for 3 and 4. The multiples of 3 are 0, 3, 6, 9, 12, 15, 18, 21 and 24 and the multiples of 4 are 0, 4, 8, 12, 16, 20 and 24. We look and see that the highlighted numbers are the ones that each 3 and 4 have in common, sinces we can not choose zero the next number that both have in common are 12 so the LCM of 3 and 4 would be 12

Primes and Factors

Prime numbers- A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. A couple of examples of prime numbers would be, 1,2,5,7,13 and so on and so forth. By using The Sieve of Eratosthenes we can eliminate all the numbers between 1-100 that are not primes. The first step is to eliminate all the numbers that end with an even number because we know that they are divisible by 2 but don't eliminate 2 so far because they are prime, then next we have to eliminate all multiples of a number (except itself). In the end we wind up with about 25 prime numbers between 1-100. You could all take the square root of the highest number that you have and the square root of that number check all the numbers below it and then the rest would be primes.


What I thought when going through my math book was the historical highlight about the Pythagorean, who were a brotherhood f mathematicians and philosophers who believed that numbers have special meanings that could account for all aspects of life. For more on the Pythagoras the founder click here. So the number theory is the study of nonzero whole numbers and their relationships. One important type of relationship in number theory is that between a factor and a multiple. If one number is a factor of a second number or divides the second then the second number is a multiple of the first.

For more examples of factoring and multiples click here. Here is a website that would be easy for children to understand factors and multiples better.

The Divisibility Test

When given a number and asked to find out what all divides into that number can be challenging and even more challenging the higher the number you go. Last week on Monday we learn the Divisibility Tests for the numbers 2-12. We wouldn't use 1 because everything is divisible by 1.This method is way easier than writing all the division out by hand, which I have done many time until my hand would cramp up. Especially on a test when you forget your calculator and home and the other calculators are in use by other students and you are stuck writing them all by hand.

When our class got to the number 7, our Teacher told us that the rule is so complex that it is easier to do the division. Test 11 on Divisibility was the most confusing one for me to do. Divisibility by Eleven: Find the sum of the odd numbered digits (odd sum) and the sum of the even numbered digits (even sum). Take the difference between odd sum and even sum. If this difference is divisible by 11, the the original number is divisible by eleven. An example would be the number 7234952. So first take the odd digit numbers 7+3+9+2=21 and even digit numbers 2+4+5=11 and subtract the two numbers 21-11=10, Ten is not divisible by eleven so the rule would not apply here. Rules 2,3,4,5,6,8,9,10 and 12 can be found here.

The Properties

There are many properties when it comes to math. The higher in math that you go, the more properties that you add on to what you know already. There are many properties for numbers such as ones for multiplication, division, addition and subtractions. There are many more once you start to add variable to your problems and equations. Here are four of the basic properties for math. Closure Property for Addition, Identity Property for Addition, Associative Property for Addition and Commutative Property or Addition.

Closure Property-For every pair of numbers in a given set, if an operation is performed, and the result is also a number in the set, the set is said to be closed for the operation. An example is for real numbers: 2, 5 are real number, 2+5=7, another real number so the set is closed for the operation. The Identity Property for Addition if For any whole number b, and the 0 is a unique identity for addition. Example 0+b=b+0=b. For the Associative Property for Addition- for any whole numbers a,b,c, a+(b+c)=(a+b)+c and for the Commutative Property for Addition- for any whole numbers a and b, a+b=b+a

Add or Subtract with Base 5

I remember when I was little I use to play with the 3-D little cubes called units, the rods of 5 units together called longs, and the big square made up of 5 longs, or 25 units, and we would use them to add and subtract with Base 5 Thinking back, it has had to have been at least over 12 years ago. I loved to make little piles or stack them up one on top of another. We would do various games that taught us math without even realizing it. Writing numbers down on a paper and using visuals are completely different.

When adding with Base 5 you cannot have over 5 or more unit pieces because you are able to switch every 5 units for a long and every 5 longs for a flat. And when subtracting with base 5, you will need to do more exchanging, exchanging the higher numbers such as flats in to longs in to units or until you get what ever number you need to satisfy. You can also use money instead because while adding or subtracting money, you can exchange little coins values or larger ones or bills and while you subtract you take the bills and break them in to coin values.

Complex to us but easy to others

There are many number systems throughout the world that many people use. You have the Quipu which is an Inca counting system, Greek, Mayan, Egyptian and Ancient Egypt counting system. Some that we know more about would be the Decimal, Base 5 and Base 12 counting system. During class we looked at six different counting systems, to me they were quite easy to catch on to because even though math isn't my favorite subject I understand and can figure word and number story problems out with hardly any trouble. Even though others struggled in the beginning because it was so new to our class, within a few tries with problems of addition and subtraction everyone understood each number system better.

Throughout class that day, Classmates were saying how confusing the Egyptian, Babylonian and Mayan counting system. Our Teacher told us that to others using the Egyptian, Babylonian and Mayan counting system, that our way looked just as confusing. In each counting system that we learned, we found the Base, if it was Positional, the Symbols they used and individual numbers represented.Represented here are the Mayan, Babylonian and Egyptian counting system and symbols.