138 and Divisibility Tricks 4 You

138 is a composite number. Factor pairs: 138 = 1 x 138, 2 x 69, 3 x 46, or 6 x 23. Factors of 138: 1, 2, 3, 6, 23, 46, 69, 138. Prime factorization: 138 = 2 x 3 x 23.

138 is never a clue in the FIND THE FACTORS puzzles.


After you learned some basic division facts, you probably realized:

  • 2 will divide evenly into any EVEN whole number.
  • 5 will divide evenly into whole numbers ending in 0 or 5.
  • 10 will divide evenly into whole numbers ending in 0.

These three rules are related to each other. All of them are true because we use base ten in our numbering system, and the prime factorization of 10 is 2 x 5.

If you needed to find the factors of a 33-digit whole number, you would be able to tell if 2, 5, or 10 divide evenly into it  just by looking at the last digit. 33-digits is more than a standard calculator can handle, but no matter how many digits a whole number has, as long as you can see the very last one, you can apply those three simple divisibility rules to know if 2, 5, or 10 are factors. Thus you will be able to do something a calculator can’t.

But wait, there are even more divisibility tricks if you can see the last TWO digits of the whole number!

divide by 4

  • 10 squared, better known as 100, divides evenly into any whole number ending in 00.
  • 5 x 5 = 25 which divides evenly into any whole number ending in 00, 25, 50, or 75.
  • 2^2 (AKA 4) divides evenly into a whole number if the final two digits can be divided evenly by 4.

How can one tell if the last two digits of a whole number are divisible by 4 (without actually dividing by 4)? I’ll show you how: I’ve put the 25 possible 2-digit multiples of 4 into one of two lists:

  • 00, 04, 08, 20, 24, 28, 40, 44, 48, 60, 64, 68, 80, 84, 88
  • 12, 16, 32, 36, 52, 56, 72, 76, 92, 96

Notice in the first list ALL the digits are even and the last digit (0, 4, or 8) can be divided evenly by 4.

Then look at the second list. The first digit is always odd and the last digit is either 2 or 6 (the only two even digits that are not divisible by 4).

Hmm. I think we can rewrite the divisibility rule for 4:

  • 4 (AKA 2^2) divides evenly into a whole number if the last two digits are even and the final digit is divisible by 4 (the last digit is 0, 4, or 8).
  • 4 divides evenly into any whole number whose next to the last digit is odd if the final digit is even but not divisible by 4 (the last digit is 2 or 6).

The rewritten divisibility rule is longer to read but takes a little less time to implement so you will have to decide which version of the rule works best for you. Either trick takes much less time than dividing some really long whole number by 4 or dividing by 2 twice.

Now I’m on to thinking about what the last THREE digits tell us.

10 Just Like Sudoku?

10 is a composite number. 10 = 1 x 10 or 2 x 5. Factors of 10: 1, 2, 5, 10. Prime factorization: 10 = 2 x 5.

When 10 is a clue in the FIND THE FACTORS puzzles, either 1 x 10 or else 2 x 5 will work for that particular puzzle.

On numerous occasions when I have demonstrated how to solve a Find the Factors puzzle, someone will remark that the puzzle is just like Sudoku. What are common factors that both a Find the Factors puzzle and a Sudoku puzzle will have? 1) Both will have only one solution. 2) Both require the solver to be able to count, write, and place the numerals 1 to 9, but Find the Factors also requires the number 10 be placed. 3) Both were originally designed to require logic to be solved. 4) Both puzzles utilize a square grid. 5) Both puzzles have several difficulty levels and variations that make the puzzles more challenging.

What factors do the two puzzles NOT have in common? 1) A difficult Sudoku puzzle can take some people almost an hour to solve while Find the Factors  would never take that long. 2) Sudoku has been a wildly popular puzzle while Find the Factors is known only among a small circle of people who have had some kind of contact with me. 3) Some more recent Sudoku puzzles require the solvers to guess and check which is getting away from its logic puzzle roots and is making it less popular for some people.  4) Sudoku could just as easily be made with letters of an alphabet, colors, or the names of the planets (if you include Pluto), while Find the Factors has to be made with numbers. 5) Sudoku requires only counting, while Find the Factors also requires the solver to factor and multiply. So really, if Find the Factors were just like Sudoku, it would look like this:


Requiring skip counting to solve the Skipoku puzzle does make it more challenging, but I became annoyed with the skip counting by the time I finished the puzzle.

One complaint about some advanced Sudoku puzzles is the need to guess and check to find a solution. Is it necessary to guess and check all the possibilities to solve this level SIX Find the Factors puzzle?


Click 10 Factors 2013-11-25 for more puzzles.

No. Guessing and checking is not necessary even though the clues 12 and 24 have several common factors.  We easily eliminate 1 and 2 because both of them would require a partner greater than 10. We also eliminate 12 because it is greater than 10. What about 3, 4, and 6? Do we have to try each of those possibilities? When we examine the puzzle we notice that it has 10 clues with only two of the clues paired together. We also notice there is one column that contains no clue, so supposedly any factor could fit there. Here is a chart of all the possible factors and the clues each could satisfy.

factors for 2013-11-25

Remember that each factor must be written twice, once in the factor row and one in the factor column. Notice that the number 9 is a factor of only one of the clues. That means that 9 has to be put over the column with no clues. From there it is easy to know where the other 9 goes and both 8’s and so forth until it is completed. Not all level SIX puzzles can be completed that easily, but using logic instead of guessing and checking is the key to solving these puzzles.