Finding Multiples & Factors of 180 & 6

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In summary, to find the number of positive integers that are multiples of 6 and factors of 180, one can consider the prime factorization of 180 and count the possibilities where the factors are multiples of 6. This can be done by taking the prime factors 2, 3, and 5 and creating permutations of them to form multiples of 6 that are also factors of 180. Another approach is to consider the fact that if 6n is a factor of 180, then 180/6n is a whole number. This can help in determining the number of possible factors. However, it is important to avoid giving the complete answer and only provide hints when assisting others.
  • #1
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How many positive integers are multiples of 6 and factors of 180?

I can find all factors of 180 and all multiples of 6 and compare, but that is not the mathematical way. I don't have a concrete idea how to start so any assistance would be more than enough.

Thanx.
 
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  • #2
Perhaps a "mathematical way" would be to consider the prime factorization, [itex]180 = 2^2 \cdot 3^2 \cdot 5 [/itex]. Therefore any factor of 180 which is a multiple of 6 is of the form [itex] 6 \cdot 2^a \cdot 3^b \cdot 5^c [/itex] where [itex] 0 \le a,b,c \le 1 [/itex]. Now it is easy to count how many possibilities there are.
 
  • #3
We know that the prime factors of 180 = 2*2*3*3*5.
So, we can take (2*3) *2 = 6*2 =12 which is a multiple of 6 and a factor of 180.
Next, take the prime factors 2*3 * 3 = 6*3 = 18. Thus, it is a multiple of 6 and a factor of 18.
Continue, you see a pattern i.e. (2*3)*5, (2*3)*3*3... of permutations.
 
  • #4
Petkovsky said:
How many positive integers are multiples of 6 and factors of 180?

I can find all factors of 180 and all multiples of 6 and compare, but that is not the mathematical way. I don't have a concrete idea how to start so any assistance would be more than enough.

Thanx.

Hi Petkovsky! :smile:

Hint: if 6n is a factor of 180,

that means that 180/6n is a whole number,

and so … ? :smile:
 
  • #5
Thank you :)
 
  • #6
Physicsissuef, don't give the answer!

(delete it if you still can)

(i'm not sure it's right anyway …)

Just give hints. :smile:
 
  • #7
6*2*3*5

Yes, my answer wasn't correct actually :D

6*2, 6*3, 6*5

Maybe this is correct.
 
  • #8
Physicsissuef said:
6*2*3*5

Yes, my answer wasn't correct actually :D

6*2, 6*3, 6*5

Maybe this is correct.

Physicsissuef, what part of "don't give the answer!" was not clear? :rolleyes:

You know we're only supposed to give help, not complete answers!

Well … you're obviously just guessing now … so no harm done! :smile:
 

What is a multiple?

A multiple is a number that can be divided evenly by another number without leaving a remainder.

What is a factor?

A factor is a number that divides evenly into another number without leaving a remainder.

How do I find the multiples of a number?

To find the multiples of a number, start by listing the counting numbers and then multiply them by the given number. For example, to find the multiples of 6, start with 1 and multiply by 6 to get 6, then 2 x 6 = 12, 3 x 6 = 18, and so on.

How do I find the factors of a number?

To find the factors of a number, start by listing all the numbers that divide evenly into the given number. For example, to find the factors of 180, you can start with 1 and 180, then 2 and 90, 3 and 60, and so on.

What are common multiples and common factors?

Common multiples are numbers that are multiples of two or more given numbers. Common factors are numbers that are factors of two or more given numbers. For example, the common multiples of 6 and 9 are 18, 36, 54, etc. The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

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