Today my blog has had 42 referral views from twitter, mostly thanks to other people’s tweets. Thank you for the tweets and retweets. Also, welcome everyone who visits my blog because of them or for any other reason!

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

Let’s try some divisibility tricks on the number 429:

4 + 2 is 6, a multiple of three, so **429** is a divisible by three. Why didn’t I include 9 in the sum? Because I already know that 9 is a multiple of three, so it is redundant to include it!

Now try another divisibility trick on **4**2**9**: I’ve put the odd numbered digits in bold and the even numbered digits in regular type. If we sum the odd numbered digits and subtract the even numbered digit, we get **4** + **9** – 2 = 11, a multiple of 11. That means that 429 is also divisible by 11!

- 429 is a composite number.
- Prime factorization: 429 = 3 x 11 x 13
- 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 429 has exactly 8 factors.
- Factors of 429: 1, 3, 11, 13, 33, 39, 143, 429
- Factor pairs: 429 = 1 x 429, 3 x 143, 11 x 39, or 13 x 33
- 429 has no square factors that allow its square root to be simplified. √429 ≈ 20.7123

**A Logical Approach to solve a FIND THE FACTORS puzzle**: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row). ** Because this is a level three puzzle**, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

That divisibility test for 11 is like doing a mod 11 operation.

Yes, it is. Then again, couldn’t you say that for any other divisibility test?

Of course.

I just remembered a trick for mentally calculating the cube roots of 3 digit perfect cubes. It also used this process.

Hmm. I don’t think I know that one. It sounds like a good trick to put up on your mathemagical site.

Maybe later since it’s quite complicated. I need to teach how to extract cube roots of perfect 2 digit cubes first, which is pretty easy.

This was such a fun post to read. I had to read the paragraph that starts with “4 +2 is 6” several times before I realized what you were saying. I just so automatically add up all the digits that it was an unexpected surprise to realize that, of course, I don’t have to add up the digits that are already multiples of 3. It seems almost silly that I didn’t realize that before!

Loved the 11 trick, though I don’t know if I’ll remember it. Maybe I will just check back in 11 days and see if you say it again.

I’m really glad you found the post fun!

The more you see or use that 11 trick, the easier it will be to remember. I promise to use it again!

42 referrals from Twitter is really nice!

Yes, it is! I’m not 100% sure, but it might be the best I’ve ever done in a day. Most days I’m lucky to get even one or two referral views from twitter.

Hmm. Some of us would be pleased to get 42 readers in a week!

I know what that’s like, too. You have posted some really helpful posts on established1962. I do hope more people will take a look at them!