A Multiplication Based Logic Puzzle

243 and Level 5

• 243 is a composite number.
• Prime factorization: 243 = 3 x 3 x 3 x 3 x 3, which can be written 243 = (3^5)
• The exponent in the prime factorization is 5. Adding one we get (5 + 1) = 6. Therefore 243 has 6 factors.
• Factors of 243: 1, 3, 9, 27, 81, 243
• Factor pairs: 243 = 1 x 243, 3 x 81, or 9 x 27
• Taking the factor pair with the largest square number factor, we get √243 = (√3)(√81) = 9√3 ≈ 15.588

This puzzle is for my sister, who is great at completing these Level 5 puzzles. Here is a black and white version for those who may prefer it:

Print the puzzles or type the factors on this excel file: 12 Factors 2014-09-22

6 A Piece of Cake

6 is a composite number. 6 = 1 x 6 or 2 x 3. Factors of 6: 1, 2, 3, 6. Prime factorization: 2 x 3.

When 6 is a clue in the FIND THE FACTORS  puzzle, the pair that will work for that particular puzzle might be 1 x 6, or it might be 2 x 3.

Scroll down the page to see the factors of 48, 720, and 1100.

In this post I talk about factor trees, and I describe how to use the cake method to find the factors of a number.

Happy birthday, Kathy! I hope your day is wonderful. You have grown into a beautiful, talented, prayerful, intelligent, hard-working, and loving young woman.  I am grateful you are my daughter.  So for your birthday today and for this blog, I’ve created three special puzzles: the first is a birthday cake to celebrate your happy day. To highlight your love of music, the second puzzle is a quarter note. The third puzzle is either a violin, a guitar, or a ukulele, you decide. I love listening as you sing or as you play any of those instruments or the piano. Today for your birthday I will also cut down a tree and make yet another cake with two birthday candles on top in this blog post.  So have a fun birthday, today.  I love you.

Click 12 Factors 2013-11-14 for more puzzles.

What did I mean by cutting down a tree and making yet another cake?  Today I will discuss two methods for finding the prime factors of a whole number.  One method is making a factor tree and the other is the cake method.  To factor a number means to write it as a product of two or more factors. When those two or more factors are all prime factors, it is called the prime factorization of the number. A composite number always has more than two factors. A prime number always has exactly 2 factors, 1 and itself. (ZERO and ONE are neither prime or composite numbers.) Usually to find the prime factors of a number, a person will usually make a factor tree.   The following example shows how this is done:

From this example, you can certainly understand why this algorithm is called a factor tree.  It looks exactly like a perfectly-shaped evergreen tree.  The problem is that a factor tree doesn’t always look so neat and trim.  Here is a factor tree that even Charlie Brown wouldn’t choose:

720 isn’t even that big of a number, but gathering all of the prime numbers from the factor tree and putting them in numerical order would be like picking up a bunch of scattered leaves. It would be like doing . . . yard work.  Imagine if you had a number that had many more factors. If one or two of the factors gets lost in the mess, your answer wouldn’t be correct. Notice that some of the prime factors of 720 (2,2,2,2,3,3,5) are not as easy to see as others on the factor tree.  That is why I want to chop down that tree. Even if you like to do yard work, do you really want to deal with that big of a mess, . . . especially when you can have cake instead?  Look, the cake method is so much more pleasing to the eye, and it is simply an extension of the very familiar division algorithm:

With the cake method, the more factors you have, the bigger the cake will be, but it will always be neatly organized with all the factors on the outside of the cake.  And if the largest prime factor of your given number is eleven, you will also have two candles on top of your cake!  I find using the cake method to be much less confusing than using a factor tree.  Yes, finding prime factors can actually be a piece of cake. The only disadvantage to the cake method is that since you work from the bottom up you have to leave enough space for the cake to rise.

Still, in spite of my opinion, it is best to use whichever method you are more comfortable with.

Now if your appetite for cake has not been satisfied, click on the amazing Spider-Man Cake or one of the other links.  Enjoy!