Number Theory

Note: This is NOT the video lesson page. If you are trying to watch a video lesson for homework, click here: https://sites.google.com/a/kcsd96.org/mrs-didech/lessons/video-lessons

We will be focusing on positive integers (1, 2, 3, 4, 5, ...)

Divisibility Rules

1 - Every number is divisible by 1
2 - If the last digit of a number is divisible by 2, then the number is divisible by 2.
3 - If the sum of the digits of a number is divisible by 3, then the number is divisible by 3. (Can be repeated)
4 - If the last two digits of a number are divisible by 4, then the number is divisible by 4.
5 - If the last digit of a number is 0 or 5, then the number is divisible by 5.
6 - If the number is divisible by both 2 and 3, then the number is divisible by 6.
7 - Take the last digit off the number and double it. Then subtract it from the remaining digits. If that difference is divisible by 7, then the original number is divisible by 7. (Can be repeated)
- ex. 283: 28 - (2•3) = 28 - 6 = 22, which is not divisible by 7, so 283 isn't either
8 - If the last three digits of a number are divisible by 8, then the number is divisible by 8.
9 - If the sum of the digits of a number is divisible by 9, then the number is divisible by 9. (Can be repeated)
10 - If the number ends in 0, then the number is divisible by 10.
11 - Add the odd numbered digits of a number, then the even numbered digits. Subtract the smaller sum from the larger sum. If that difference is divisible by 11, then the original number is divisible by 11.
- ex. 29383: 2 + 3 + 3 = 8. 9 + 3 = 12. 12-8 = 4, which is not divisible by 11, so 29383 isn't either.
12 - If a number is divisible by both 3 and 4, then the number is divisible by 12.

Factors

A number that divides another number with no remainder . For example, 12 has six factors: 1, 2, 3, 4, 6, 12. On the other hand, 13 has two factors: 1, 13.

Multiple

A number that is multiplied by another number. Multiples of 3 include: 3, 6, 9, 12, 15, 18, ...

Primes and Composites

A prime number has exactly two factors: 1 and itself. Some prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Composite numbers have three or more factors. The number 1 is neither prime nor composite because it only has one factor: itself. The way to determine if a number is prime or composite is to test divisibility using the rules above. We know that whatever number we start with automatically has two factors, 1 and itself, so we only need to find one more before we can stop. Here is what this process might look like:

Is 49 prime? We know 1 and 49 are factors, but we need one more. Is 49 divisible by 2? No. How about 3? No. Does 4 work? No. 5? No. 6? No. 7? Yes! 49 is composite and we can stop.

Usually it will work if you test the numbers 2-10. If you know how to find square roots, find the square root of the original number first. Then test all of the numbers that are less than or equal to the square root, at which point you can conclude the number is prime if you haven't found a factor.

Greatest Common Factor

To find the greatest common factor of two (or more) numbers, list out all of the factors of the small number from least to greatest. Starting with the greatest factor, check if it is also a factor of the other number. If it isn't test the next one. Continue until you find a shared factor.

Least Common Multiple

To find the least common multiple, start to list multiples of each number. The least number that they share is the least common multiple.

Cake Method for GCF and LCM

Here is a good website that demonstrates our method for finding GCF and LCM:

http://bit.ly/MKzpOp

Students Teaching Students

Nathan and Logan (Cake Method Personal Video)

Christine and Allison (Number Theory Slide Show)

Hannah (GCF and LCM Video Lesson)

Resources:

Divisibility Rules Games

http://www.basic-mathematics.com/divisibility-rules-game.html