A Multiplication Based Logic Puzzle

Posts tagged ‘5’

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.

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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.

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5 Easy as 1-2-3

5 is a prime number. 5 = 1 x 5. Its only factors are 1 and 5. Prime factorization: none.

How do we know that 5 is a prime number? The square root of 5 is an irrational number approximately equal to 2.24. If 5 were not a prime number, then it would be divisible by at least one prime number less than or equal to 2.24. Since 5 is not divisible by 2, it is a prime number.

When 5 is a clue in a FIND THE FACTORS puzzle, use 1 for one of the factors and 5 for the other.

Being able to identify factors of a whole number is a very important skill in mathematics.  It is a skill that is commonly used in many areas of mathematics ranging from reducing fractions to solving differential equations.  The Find the Factors puzzles can help make that skill second nature.

2013-11-11

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

To solve the puzzles, we are only interested in the limited set of factors that are represented in the following table:

Puzzle Clues Chart

What about all the other factors of these numbers?  And what about all the other whole numbers not on the chart?  How do you find ALL of the factors of a given whole number?  For example, suppose you were asked to find all of the factors of 435.  Some people might notice right away that it is divisible by 5 because its last digit is 5.  While that is true, beginning with 5 is not the best place to start because there is an advantage in considering all possible factors in an organized way.  When you are asked to find ALL of the factors of any number, starting at 1 will make finding all of the factors as easy as 1-2-3. So what are the factors of 435?  Using a calculator, I notice that the square root of 435 is about 20.85.  That means I can find absolutely all of the factors of 435 by considering as  divisors just the whole numbers from 1 to 20!  Each factor will have a partner that is greater than 20 but will be found at the same time with these few short calculations. To demonstrate my thinking process, I will put each possible factor from 1 to 20 in a chart and write my thoughts as I consider each one.

Thinking part 1

Thinking part 2

As you may notice, once a possible factor is eliminated, it is not necessary to do any actual division by ANY of the multiples of that number. (4, 6, 8, 10, 12, 14, 16, 18, and 20 are all multiples of 2, which was not a factor, so I didn’t actually divide 435 by any of those multiples.)

As I carefully consider each possible factor, I only WRITE DOWN a number if it is an actual factor.  Therefore, with only a little bit of effort I would list ALL of the factors of 435 in one tidy list: 1 x 435, 3 x 145, 5 x 87, 15 x 29.

See, it was as easy as 1-2-3!  Now let’s find all of the factors of 144.

factors of 144

Even though 144 is less than 435, it has more factors. One of its factors is paired with itself because the square root of 144  is 12.  That fact is also the signal that we can stop looking for more factors, and we can list all the factors of 144 on the following chart:

144 table

There are 8 multiplication facts that produce 144, but 12 x 12 = 144 is the only fact we consider when solving a Find the Factors 1-12 puzzle with 144 as one of the clues. In every other case one of the pair of numbers in the multiplication fact will be greater than 12 and not eligible to be written in the factor row or factor column. However in solving mathematical problems, any of the factors of a whole number could be the star of the show. Knowing how to find those factors is indeed an important skill and is as easy as 1-2-3.

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