## 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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