## Prime Factorization: When Students Ask

• Why should I bother learning this?
The prime factorization of a number is used in many algorithms such as finding the least common multiple and the greatest common divisor. These in turn are used in working with fractions. The least common multiple is used when finding the lowest common denominator, and the greatest common factor is used in simplifying a fraction. Many patterns, formulas, and number concepts in number theory rely on the ability to express a number as a product of prime numbers. For example, a perfect number is one whose proper factors (factors less than the number) add up to the given number. The smallest perfect number is six, and its proper factors are 1, 2 and 3. After showing that six is perfect, you could ask students to find the next perfect number (28).
• What is the greatest prime number?
There is no greatest prime number. The greatest prime number discovered so far has 895,932 digits, but there are undoubtedly greater ones. A famous mathematician named Euclid was able to prove many years ago that there is no greatest prime number.
• Are there rules for divisibility for 6, 7, 8, and 11?
Yes, there are rules for dividing by the numbers 6, 7, 8, and 11. For example, six will divide a number if it is even and the sum of the digits is divisible by three. The rules for dividing by 7, 8, and 11 are more complex to follow than is to do the division itself.