# Pre-Algebra:

I am working on least common multiple, greatest common divisor, and divisors. I already posted the rules for least common multiple, so here are the other rules for the last two:

**Important**: if a and b are nonzero integers, then all of the following statements mean the same thing:

1. A is a divisor of b.

2. A is a factor of b.

3. B is a multiple of a.

4. B is divisible by a.

**Important**: if a is a divisor of b, then every divisor of a is also a divisor of b.

**Important**: let a, b, and c be integers. If c is a divisor of a and b, then c is a divisor of a + b and of a – b.

**Important**: Let n be a positive integer. The prime factorization of any multiple of n includes the prime factorization of n. that is, every prime in the prime factorization of n is in the prime factorization of every multiple of n, and is raised to at least as great power in the multiple at it is in n.

**Important**: Let n be a positive integer. The prime factorization of n includes the prime factorization of each divisor of n. that is, every prime in the prime factorization of each divisor of n is in the prime factorization of n, and is raised to no greater a power in the divisor than it is in n.

**Important**: we can find the greatest common divisor of a group of numbers with the following process:

**Step 1**: Find the prime factorization of each number.

**Step 2**: Identify the primes that appear in all of the prime factorizations.

**Step 3**: Among the prime factorizations, find the smallest power of each prime from step 2.

**Step 4**: Multiply the powers of primes found in step 3.

**Important**: if two integers do not have any prime factors in common then the integers are relatively prime.

**Important**: Let a and b be integers. Every common divisor of a and b is a divisor of gcd(a, b,), and every divisor of gcd(a, b,) is a common divisor of a and b.

**WARNING**!! A prime must appear in the prime factorization of all of the numbers in a group in order to appear in the prime factorization of the greatest common divisor of the group.

**Important**: For any positive integers a, b, and n, we have

Gcd(n · a, n · b) = n · gcd(a, b)

**Important**: To test for divisibility by a composite number, we can test for divisibility by two relatively prime numbers whose product is the composite number.

# Java Programming:

In Youth Digital, I learned how to create my own biome. Surprisingly, it was simple. there was one section that would generate things like plants, trees and other natural things like that. to show this to you, here is my code from eclipse:

this.theBiomeDecorator.flowersPerChunk = 27;

this.theBiomeDecorator.treesPerChunk = 3;

this.theBiomeDecorator.bigMushroomsPerChunk = 1;

this.theBiomeDecorator.cactiPerChunk = 2;

this.theBiomeDecorator.grassPerChunk = 25;

this.theBiomeDecorator.reedsPerChunk = 10;

this.theBiomeDecorator.sandPerChunk = 5;

this.theBiomeDecorator.sandPerChunk2 = 5;

this.theBiomeDecorator.clayPerChunk = 30;

“theBiomeDecorator” Is telling minecraft that in my biome these decorations will be loaded in my biome. The “.____” is telling minecraft what is being loaded. The “PerChunck = __” is saying “in every 16 x 16 area ___ will try to spawn.” somethings like clay have to spawn under rivers. so the only time clay spawns in in a 16 x 16 area that has water, and then the clay is spawned underneath that river or ocean. you can also customize what mobs spawn in that biome. this is my code:

this.spawnableCreatureList.add(new SpawnListEntry(EntityChicken.class, 12, 4, 8));

this.spawnableCreatureList.add(new SpawnListEntry(EntityCow.class, 12, 4, 8));

his.spawnableCreatureList.add(new SpawnListEntry(EntityHorse.class, 12, 4, 8));

this.spawnableCreatureList.add(new SpawnListEntry(EntitySheep.class, 12, 4, 8));

this.spawnableCreatureList.add(new SpawnListEntry(EntityPig.class, 12, 4, 8));

this.spawnableCreatureList.add(new SpawnListEntry(EntityOcelot.class, 12, 4, 8));

his.spawnableCreatureList.add(new SpawnListEntry(EntityWolf.class, 12, 4, 8));

this.spawnableCreatureList.add(new SpawnListEntry(EntityIronGolem.class, 5, 4, 8));

The “spawnableCreatureList” tells minecraft the list of things that can spawn. The “.add” is saying that the code after it is being added in the “new SpawnListEntry” and its adding entity chicken, or entity Pig. The “.class, __, __, __” is telling minecraft how common it is to spawn, the least amount that can spawn in a group, and the biggest amount that can spawn in a group. also in my code that is worth explaining:

this.setMinMaxHeight(0.6F, 0.6F);

this.setTemperatureRainfall(1.0F, 0.3F);

the “.setMinMaxHeight” and the “.setTemperatureRainfall” is basically a key for the parentheses. the first number is the Min Height, and the other one is the average Temperature. And the other ones are just the second word! it is really incredible how much you can do in so little time!

# Ancient Civilization:

I finished my Minecraft Homeschool Castles & Cannon course. **I made a 97 in the class! **I'm taking engineering and science in the Redstone Academy next session! I'm very excited! Here are my grades and then info on my Redstone Academy course:

I can’t wait to see the biomes you created!!!

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Nice! We can’t wait to see your final world!

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