14 is the GCF of a number \( M \) and 210. What are the possible values of \( M \)?

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, .............

Answered by rabinshrestha41 | 2024-06-10

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.

Answered by Illuminati750 | 2024-06-10

The possible values of M include multiples of 14, such as 14, 42, 70, and more, which maintain the GCF of 14 with 210. To find M , note that it must have 2 and 7 as factors alongside possible additional primes from 210. This means M can be expressed as M = 14 k , where k is an integer that incorporates factors of 3 and 5 as necessary.
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Answered by rabinshrestha41 | 2024-10-01