2008-Feb–23 Note:

* One of the most common methods of teaching factoring is using a graphic tool called a factor tree.  Factor lines are not only intuitively superior to factor trees but are environmentally superior due the reduced space to perform the factoring. Insert powerpoint presentation here but provide general preview for early browsers.]

Factor Lines* are derived from simple extension of the divisor line into a continuous line.  The results of the previous division creates the next numerator, the smallest divisor creates the next denominator.  An example should suffice for now.

16380  8190  4095  1365  455   91   13    1 (we’re done)

16380 =  1 *  2    *    2   *   3    *   3  *    5  *  7  * 13

All digits are in base A (ten)

- Division by 1 is recursively infinite so 1 initiates the denominator and only occurs once.

- Division by 2 occurs for 50% of all numbers. Check for evenness.

- Division by 3 occurs for 33 1/3%.  Verify that the sum of digits is divisible by 3.

- Division by 5 occurs for 20% of all numbers. Verify that the LSD ends in 0 or 5.

- Division by 7. Double LSD and subtract from remaining MSD's is divisible by 7.

- Division by 11. Difference between the sums of the even and odd digits is divisible by 11.

(LSD - least significant digit, MSD - most significant digit)

RESEARCH

see http://mathforum.org/k12/mathtips/division.tips.html for general prime case.

see http://mathforum.org/k12/mathtips/ward.html for polynomial based explanation

Provide question on why the denominator is a prime if its square is greater than the numerator.