# ## Factors and Fractions: When Students Ask

• Why do I need to know about prime numbers and prime factorization?
You can use the prime factorization of a number to find the LCM and GCF of two or more numbers. As you continue to study mathematics, you will find that many patterns, formulas, and number concepts in number theory rely on prime numbers and the ability to express a number as a product of prime numbers. For example, a mathematician named Goldbach made a conjecture that every even number greater than 4 can be written as the sum of two odd prime numbers. Examples of the Goldbach conjecture are
42 = 13 + 29, 66 = 19 + 47, and 120 = 59 + 61. This conjecture has never been proved or disproved.
• When would I use the least common multiple?
When adding fractions, you may need to find a common denominator. You can find the least common denominator by finding the LCM of the denominators of the fractions to be added. You can also use the LCM when comparing fractions, by writing them as equivalent fractions, using the LCM of the denominators.
• When do I use the greatest common factor?
A common use of the GCF is to simplify a fraction by dividing both the numerator and denominator by the GCF of both. Another way to find the LCM for two numbers is to divide the product of the two numbers by the GCF for the numbers. For example, the GCF of 36 and 60 is 12. The product of 36 x 60 = 2,160. Therefore, the LCM of 36 and 60 is 2160 ÷ 12, or 180.
• How are equivalent fractions useful?
Equivalent fractions are used many ways. They can be used to compare two fractions with unlike denominators. When adding two fractions with unlike denominators, you may need to change the fractions to equivalent fractions with the same denominators.