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14 is the GCF of a number M and 210. What are the possible valued of M?

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Illuminati750idn
Hello, 

we know that to find the GCF we descompose the numbers and then we multiply the common terms with the lowest exponent. So we first descompose 210:

210   | 2
105   | 3
  35   | 5
    7   | 7
    1   

210= 2x3x5x7

Now one possible solution is 14 because:

14   | 2
  7   | 7
  1

As you can see 2 and 7 are common terms, so:

GCF= 2x7
GCF=14   -->  It's correct

Then, if we analyze it, we'll realize that we have an infinitive number of solutions. The only requirement is that "M" must have 2 and 7 when we descompose it.

For example:

M= 2²x7x3x11
M=924

924 is another possible solution, it doesn't matter if 2 or 7 repeat, since I said the GCF is the multiplication of the common terms with the LOWEST exponent. That's all.
   
Rabinshrestha41idn
First, find all the prime factors of 210.

Prime factors of 210 are : 2, 3, 5, 7

Given, The GCF of M and 210 is 14.
And GCF is obtained by multiplying common prime factors, 

So, M must be a number with 2 and 7 as a prime factor because (2 * 7) =14

So, any number with 2 and 7 as its prime factors are possible values of M. Some of them are 70, 98, 154, .............

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