## A Multiplication Based Logic Puzzle

### 701 Some Virgács left by Mikulás

• 701 is a prime number.
• Prime factorization: 701 is prime and cannot be factored.
• The exponent of prime number 701 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 701 has exactly 2 factors.
• Factors of 701: 1, 701
• Factor pairs: 701 = 1 x 701
• 701 has no square factors that allow its square root to be simplified. √701 ≈ 26.4764046.

How do we know that 701 is a prime number? If 701 were not a prime number, then it would be divisible by at least one prime number less than or equal to √701 ≈ 26.4. Since 701 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 701 is a prime number.

Tonight Mikulás will visit the homes of children who sleep in Hungary. If they have been good, he will fill their boots with sweet treasures. If they have been naughty, they will receive virgács, small twigs that have been spray painted gold and bound together with red decorative ribbon. Actually most children make both the naughty list and the nice list so their boots are filled with a mixture of sweet and the not so sweet including virgács, a subtle reminder to be good.

I especially like this illustration from Wikipedia that features Mikulás (Saint Nickolas) and Krampusz:

I like that it is 150 years old. It is from 1865, several decades before any of my husband’s grandparents left Hungary to live in the United States. Under the chair is a little boy hiding from Krampusz. I like to imagine he’s related to my husband some way. The little girl in the illustration must have been much better behaved that year because she is not afraid enough to need to hide.

Since everyone has been at least a little bit naughty this year, here is virgács for you to put in your shoes tonight, too.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here is a little more about the number 701:

26² + 5² = 701 so it is the hypotenuse of the primitive Pythagorean triple 260-651-701 which can be calculated using 2(26)(5), 26² – 5², 26² + 5².

701 is the sum of three consecutive prime numbers: 229 + 233+ 239.

701 is a palindrome in several bases:

• 10301 BASE 5; note that 1(625) + 0(125) + 3(25) + 0(5) + 1(1) = 701.
• 858 BASE 9; note that 8(81) + 5(9) + 8(1) = 701.
• 1F1 BASE 20; note that F is equivalent to 15 in base 10, and 1(400) + 15(20) + 1(1) = 701.
• 131 BASE 25; note that 1(625) + 3(25) + 1(1) = 701

Stetson.edu informs us that 1^0 + 2^1 + 3^2 + 4^3 + 5^4 = 701.

Here’s another way we know that 701 is a prime number: Since 701 ÷ 4 has a remainder of 1, and 701 can be written as the sum of two squares that have no common prime factors (26² + 5² = 701), then 701 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √701 ≈ 26.4. Since 701 is not divisible by 5, 13, or 17, we know that 701 is a prime number.

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### 317 and Candies from Mikulás

• 317 is a prime number.
• Prime factorization: 317 is prime and cannot be factored.
• The exponent of prime number 317 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 317 has exactly 2 factors.
• Factors of 317: 1, 317
• Factor pairs: 317 = 1 x 317
• 317 has no square factors that allow its square root to be simplified. √317 ≈ 17.804

How do we know that 317 is a prime number? If 317 were not a prime number, then it would be divisible by at least one prime number less than or equal to √317 ≈ 17.804. Since 317 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 317 is a prime number.

Today is Mikulás or St. Nickolas’s name day in Hungary. Children woke up to find various candies such as the candy cane below in their shined shoes this morning.

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01

My grandson left this note to Mikulás last night.

Köszönöm Mikulás means “Thank you, St. Nickolas.” My grandson was thrilled with the candies and amused by the virgács that he found in his shoes this Mikulás morning .

### 312 and Virgács

Today might not be the 312th day of the year, but here’s a fun fact about the number 312:

• 312 is a composite number.
• Prime factorization: 312 = 2 x 2 x 2 x 3 x 13, which can be written 312 = (2^3) x 3 x 13
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 312 has exactly 16 factors.
• Factors of 312: 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312
• Factor pairs: 312 = 1 x 312, 2 x 156, 3 x 104, 4 x 78, 6 x 52, 8 x 39, 12 x 26, or 13 x 24
• Taking the factor pair with the largest square number factor, we get √312 = (√4)(√78) = 2√78 ≈ 17.664

The night of December 5th, Mikulás, or St. Nickolas, will visit homes in Hungary, some neighboring countries, and even a few houses in the United States.  Children who have been good will awaken to find their shoes filled with little treats such as candy, fruit or nuts. Since all children are occasionally a little bit naughty, they will also find virgács, a small collection of twigs that have been spray-painted gold and decoratively bound together.  Virgács are not sold in the United States, so St Nickolas will be making some himself using the bristles from a natural broom.  The finished product should look like this:

This puzzle is meant to be reminiscent of the virgács my children’s ancestors probably received each year on St Nickolas Day.
Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01
A Logical Approach to FIND THE FACTORS: 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.

### 13 Mikulás

13 is a prime number. 13 = 1 x 13. Its only factors are 1 and 13. Prime factorization: none.

How do we know that 13 is a prime number? The square root of 13 is an irrational number approximately equal to 3.606. If 13 were not a prime number, then it would be divisible by at least one prime number less than or equal to 3.606. Since 13 is not divisible by 2 or 3, it is a prime number.

13 is never a factor in the FIND THE FACTOR 1 – 10 or 1 – 12 puzzles.

Tonight (December 5th) all over Hungary, children will polish their boots and then place them in a window or in front of their door. Once the children are “nestled, all snug in their beds, … visions of sugar-plums (will indeed) dance in their heads” as they await a visit from Mikulás, or St. Nickolas.  When the children get up in the morning, they will find their boots or shoes filled with candy, fruit, and nuts if they have been good. If they have been bad, their boots or shoes will be filled with virgács, a small collection of twigs that have been spray-painted gold and decoratively bound together.

Since most children were good some of the time and naughty once in awhile, they will likely find  the expected goodies as well as virgács in their shoes or boots.
With these traditions in mind, I created the puzzles for today. If you have just a little imagination, you will be able to see different types of candy as well as the virgács in the clues. These puzzles will be a treat to any child or adult who did their homework and learned multiplication, division, and factoring. Click 12 Factors 2013-12-05.