Question

Your cousin, who is in middle school, is learning about prime numbers and prime factorization for the first time, and is struggling with the concept. 1. Explain to your cousin, in a way that would make sense to a middle school student, what it means for a number to be prime. Use this to explain why 1 is not a prime number, and to explain what the first four prime numbers are. 2. Next, explain what it means to find the prime factorization of a number. Then demonstrate your idea of prime factorization by trying to write the numbers 2 through 20 as a product of any combination of the first four prime numbers

Answer 1

Do you have any ideas? Can you tell me what a prime number is to start if you don't have any ideas?

Answer 2

I understand the first question i just don’t get the second one

Answer 3

Ok, so can you please answer the first part for me then so I can get a sense of your understanding?

Answer 4

prime numbers are the numbers that are bigger than 1 and cannot be divided evenly by any other number except 1 and itself

Answer 5

1 is not considered a prime because it has one factor

Answer 6

good, and why isn't 1 one of them? It's divisible by 1 and itself?

Answer 7

good good

Answer 8

prime numbers have two factors

Answer 9

ex: the prime factors of 12 are 2x2x3

Answer 10

the first 4 prime factors are 2,3,5,7

Answer 11

wait

Answer 12

are you there?

Answer 13

prime numbers have 2 factors? then your ex of 12. 12 is not prime. Type-o?

Answer 14

well my second question is about prime factorization

Answer 15

prime factors are multiplied together to make the original number ex :12=2x2x3

Answer 16

ok, so conceptually close but not entirely, just a quick correction: A prime factorization breaks a number up into multiplication, but the numbers you are multiplying are only prime.

Answer 17

my question is writing the numbers 2 through 20 as a product of any combination of the 1st four prime numbers? i have no idea what this means

Answer 18

do you see the difference on the lead in? then your ex of 12 where you have 12=2^2*3 would make more sense

Answer 19

so do i have to write numbers from 2 to 20 and break them up into multiplication?

Answer 20

like 8=2^3 9=3^2 and so on until 20?

Answer 21

And as for your question, it's a concept you are missing, that is all. the prime factorization is not just 2 primes, it is actually \[n= p_{1}^{a_1}* p_{2}^{a_2}* p_{3}^{a_3}...\]

Answer 22

the p are your different primes, and the a are how many times they appear in your factorization.

Answer 23

yes, you did the factorization for 8 and 9 correct, but what about2,3,4,5,6,7?

Answer 24

that is the part i don’t understand?

Answer 25

ok can you pin point where?

Answer 26

After i explained what it means to find the prime factorization of a number i have to demonstrate my idea of prime factorization by trying to write the numbers 2 through 20 as a product of any combination of the first four prime numbers.

Answer 27

yes i understand, but i need to know exactly where your confusion is

Answer 28

OH WAIT, I think I have a guess, here is an example 23 is prime right? so it's factorization is simply 23=23

Answer 29

yeah i understand that but is this question asking me to list the numbers from 2 to 20 and factor them to see if i get any combination of the first 4 prime numbers?

Answer 30

well, let's start writing them
2=?
3=?
4=?
5=?
6=?

Answer 31

it just wants you to list all of their prime factorizations

Answer 32

oh i see. so 2=2, 3=3,4=2^2…and so on

Answer 33

from 2-20, using just the first for primes if possible

Answer 34

yea, the hiccup will come when you hit 11(you cannot use the first four primes only because 11 is prime)

Answer 35

You will need to explain why that is

Answer 36

so what would the prime factorization for 16 be?

Answer 37

so why can’t 11, 17, 19 be factored into primes?

Answer 38

What are 11, 17, and 19? They themselves are_______.

Answer 39

16 is 2x2x2x2=2^4

Answer 40

yup good

Answer 41

(17,19) prime numbers

Answer 42

so they only have two factors?

Answer 43

good, and what do you know about primes? What are the only numbers that divide them?

Answer 44

1 and itself

Answer 45

good, so now is it possible to multiply any group of numbers and get a prime if that group does not contain the prime itself?

Answer 46

not all numbers cannot be written as a combination of the first 4 prime numbers so no

Answer 47

i also have to explain why some of these numbers cannot be factored into primes.

Answer 48

Well, yea, but basically what I'm getting at is that a prime decomp of a prime, is just that prime. Follow?

Answer 49

I left out a question by accidentYour cousin notices that some of the numbers cannot be written as a combination of the first four prime numbers. Explain why these numbers could not be factored into primes.

Answer 50

Well tell me, why can't 19 be written as a factorization of the first 4?

Answer 51

well because 2,3,5,7 are not factor of 19. 19 is its own prime factor

Answer 52

good, are there any others that are different?

Answer 53

yes (11,13,17) if i go up to 30 (23 and 29)

Answer 54

Good good, now you can explain why, so I think my work here is done. You are fully capable of answering the question. :) good work

Answer 55

thank you so much :)