Technically, the first step is to factor each of the first two numbers, to “find their prime factors“. A factor is any integer that divides another integer evenly. A prime factor is also a prime number, whose only factors are ‘1’ and itself.

Sometimes a factor is also called a “divisor”. Let’s return to finding factors.

We know “50 = 2 * 5 * 5”, while “75 = 3 * 5 * 5”. The common factors are the two ‘5’s.

To find the LCM, we only need to use the all the common factors once, and then include all the unique factors. So the LCM of ’50’ and ’75’ is “(5 * 5) * (2 * 3) = 150”.

Here is the general rule for finding the LCM for positive integers ‘a’ and ‘b’.

- Let’s say that a[i] are prime factors unique to ‘a’, b[i] are unique to ‘b’, and c[i] are common to ‘a’ and ‘b’.
- Let’s say that “a = a[1]*a[2]*…*a[j]*c[1]*c[2]*…*c[m]” and “b = b[1]*b[2]*…b[k]*c[1]*c[2]*…*c[m]”.
- Then the LCM of ‘a’ and ‘b’ is “c[1]*c[2]*…*c[m]*a[1]*a[2]*…*a[j]*b[1]*b[2]*…b[k]”.

## Vocabulary for Fractions: Simplest Terms

If the numerator and denominator of a fraction have a common factor, it’s good to simplify the fraction by dividing each by the largest common factor.

A fraction in simplest terms cannot be simplified in this manner.

An example is simpler than the explanation. “150/475 = (6 * 25)/(19 * 25) = 6/19”. The largest common factor of ‘150’ and ‘475’ is ’25’. Dividing both the numerator and denominator of “150/475” leaves “6/19”. The only common factor for ‘6’ and ’19’ is “one”, so the simplest terms for “150/475” is “6/19”.** **

## Fraction Identity and Multiplication Identity

One very useful rule for fractions, and for division in general, is that dividing any non-zero number by itself results in the value “one”. Both “3/3 = 1” and, so long as ‘b’ does not equal zero, “b/b = 1”.

A bonus rule is that ‘1’ is the identity for multiplication. That simply means that “x * 1 = x” for any ‘x’; multiplying by ‘1’ leaves ‘x’ unchanged.

Since “b/b = 1” and “x * 1 = x”, it follows that “x * (b/b) = x * 1 = x”.

## “What are Fractions?” Answered: What’s Next?

Now that you’ve learned the basics of fractions, and the vocabulary that goes with them, you’re ready to add fractions together. If you need more help or practice to learn about these math terms, ask your teacher, search for online math worksheets for fractions, ask a question in our Ask the Experts section or in the comments below, or try a math kit for learning fractions.

**References**:

Weisstein, Eric W. *Least Common Multiple* and *Mixed Fraction*. MathWorld-A Wolfram Web Resource. Referenced January 7, 2012.

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