Finding the lowest common denominator
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The LCM is used for integers, not fractions. If you're trying to add unlike fractions, take the LCM of the denominators (known in this case as the least common denominator, or LCD), convert the fractions and proceed.
GCF doesn't apply to fractions, only to whole numbers. When those numbers are the numerator and denominator of a fraction, the GCF can be used to reduce it to its simplest form.
Fractions are infinitely dense and this means that between any two fractions there an infinite number of fractions. If any fraction, f, laid claims to being the nearest, there would be infinitely many fractions between 0 and f and so infinitely many fractions which were closer to 0. This means that f could not be the closest. The argument can be used again and again and so there cannot be a fraction closest to 0.
When subtracting one fraction to another, one or both fractions are renamed so that they have the same denominators. Then the result of the subtraction is the equal to the subtraction of the numerators divided by the common denominator.