## A Multiplication Based Logic Puzzle

### A Teacher Affects Eternity and Divisibility Tricks Applied to 418

Henry Brooks Adams said, “A teacher affects eternity; he can never tell where his influence stops.” Of course the quote applies to female teachers just as much as it does to male teachers. Affecting eternity is pretty serious business.

It will probably take less than two minutes to read Marek Bennet’s  comic on how the work you do fits into the educational system. Its message is worth pondering for quite a long time.

Let me demonstrate some quick prime number divisibility tricks on the number 418:

• It’s even so its divisible by 2.
• 4 + 1 + 8 = 13, which is not a multiple of 3, so 418 is not divisible by 3.
• The last digit isn’t 0 or 5, so 418 is not divisible by 5.
• 41 – 2(8) = 41 – 16 = 25, which is not a multiple of 7, so 418 is not divisible by 7.
• If we add the first and the 3rd digits of 418, we get twelve. Then if we subtract the second digit, one, we get eleven. Since we got a multiple of eleven, that means that our original number, 418, can be evenly divided by eleven.

Here is the factoring information for 418:

• 418 is a composite number.
• Prime factorization: 418 = 2 x 11 x 19
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 418 has exactly 8 factors.
• Factors of 418: 1, 2, 11, 19, 22, 38, 209, 418
• Factor pairs: 418 = 1 x 418, 2 x 209, 11 x 38, or 19 x 22
• 418 has no square factors that allow its square root to be simplified. √418 ≈ 20.445