need help with maths

zoe2002

16

how do you do the unique factorisation theorem? I don't get it ?? please help me :smile:

Reply 1

thefatone

6

Original post by mayu123

how do you do the unique factorisation theorem? I don't get it ?? please help me :smile:

so basically just divide the number using prime numbers then write it up.

for example using the "Unique factorisation theorem" as you put it on the number 200

5 is a prime number so i can use that

200 \div 5 = 40

40 \div 5 =8

2 is a prime number too so i can use that

8 \div 2=4

4 \div 2=2

so now we count up how many times we had to divide the number 200 by prime numbers, we also need to include the last number which is 2 aswell

so we have 2 5s and 3 2s so we can write this in the form

2^3 \times 5^2 =200

Reply 2

Zacken

22

Original post by mayu123

how do you do the unique factorisation theorem? I don't get it ?? please help me :smile:

It's more commonly called the "fundamental theorem of arithmetic" - to prove this theorem, you need to make use of Euclid's Lemma. That is,

p | ab \Rightarrow p | a \vee p | b

.

Use strong induction to show that every integer greater than 1 is a product of primes. The proof is quite elementary.

The uniqueness part is slightly harder, assume an integer is a product of primes in two different ways:

a = p_1\cdots p_n = q_1 \cdots q_m

then we need to show that this is the same thing and

m=n

.

Now use Euclid's Lemma on each

p_i

and

q_i

to show that

p_i = q_i

(up to relabelling).

Reply 3

zoe2002

OP

16

Original post by thefatone

so basically just divide the number using prime numbers then write it up.

for example using the "Unique factorisation theorem" as you put it on the number 200

5 is a prime number so i can use that

200 \div 5 = 40

40 \div 5 =8

2 is a prime number too so i can use that

8 \div 2=4

4 \div 2=2

so now we count up how many times we had to divide the number 200 by prime numbers, we also need to include the last number which is 2 aswell

so we have 2 5s and 3 2s so we can write this in the form