## A Multiplication Based Logic Puzzle

### 409 Happy Birthday, John!

409 = (20^2) + (3^2) so it is the hypotenuse of a Pythagorean triple. That triple is [120, 391, 409]. Could 409 possibly be a prime number? The answer is at the end of the post.

Today is my son’s birthday. He lives on the other side of the country, so I’m making him a puzzle cake that he can devour. Many years ago when he still lived at home, he could solve any of my puzzles, so I know he can handle this one even though I haven’t revealed its difficulty level.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

I am grateful for facebook! His wife posted, “I felt like I was wrapping presents for a 10 year old boy this morning.” My son is a young father who loves, loves, loves Legos.

• 409 is a prime number.
• Prime factorization: 409 is prime and cannot be factored.
• The exponent of prime number 409 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 409 has exactly 2 factors.
• Factors of 409: 1, 409
• Factor pairs: 409 = 1 x 409
• 409 has no square factors that allow its square root to be simplified. √409 ≈ 20.2237

How do we know that 409 is a prime number? If 409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √409 ≈ 20.2237. Since 409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 409 is a prime number.

### 408 and Level 6

408 is divisible by 8 and so are 1408, 2408, 3408, 4408, . . . . . . and any other number whose last 3 digits are 408. This is true because if the last 3 digits of a number are divisible by 8, the entire number is also!

Also since 4 + 8 = 12, a multiple of 3, we know that 408 is divisible by 3.

408’s factors are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 408 is a composite number.
• Prime factorization: 408 = 2 x 2 x 2 x 3 x 17, which can be written 408 = (2^3) x 3 x 17
• 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 408 has exactly 16 factors.
• Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408
• Factor pairs: 408 = 1 x 408, 2 x 204, 3 x 136, 4 x 102, 6 x 68, 8 x 51, 12 x 34, or 17 x 24
• Taking the factor pair with the largest square number factor, we get √408 = (√4)(√102) = 2√102 ≈ 20.199

### 407 What’s the Logical Thing to Do, Spock?

I was sad to learn of the death of Leonard Nimoy today. The factors of 407 will have to wait until the end of the post.

We only had one television when I was growing up, and my brothers insisted that it be tuned into Star Trek each week. Soon enough I became as interested as they were. After Star Trek was canceled, I loved watching Leonard Nimoy on Mission Impossible. I also loved every Star Trek movie he was ever in especially #2, 3, and 4. I was delighted that he was in the J. J. Abrams versions, and it’s painful to realize he will not be in any more.

When faced with any challenge, his character, Spock, always looked for the logical thing to do. I don’t imagine he would have wasted much time doing logic puzzles such as mine. Nevertheless, I dedicate this one to him.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

My husband and I didn’t watch Fringe until its second season when we heard that Leonard Nimoy would be in it. Then we watched it every week, often learning more about his character even though he only occasionally made an appearance.

We recently loved watching him in an old episode of Columbo even though he didn’t get away with his perfect crime.

Leonard Nimoy has died at the age of 83, a prime number for a man who always seemed to be in his prime.

On numerous occasions he has touched me, and his mind actually melded a little bit into mine. Some of who he was and what he did in life will permanently be a part of me.

Now here is 407’s factoring information:

• 407 is a composite number.
• Prime factorization: 407 = 11 x 37
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 407 has exactly 4 factors.
• Factors of 407: 1, 11, 37, 407
• Factor pairs: 407 = 1 x 407 or 11 x 37
• 407 has no square factors that allow its square root to be simplified. √407 ≈ 20.1742

### 406 and Level 4

Let’s apply the 7 divisibility trick to 406.

• Separate the last digit from the rest and multiply that last digit by two: 6 x 2 = 12.
• Now find the difference between the remaining number, 40 and 12: 40 – 12 = 28.
• Since 28 is a multiple of 7, we know that 406 is also divisible by 7.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 406 is a composite number.
• Prime factorization: 406 = 2 x 7 x 29
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 406 has exactly 8 factors.
• Factors of 406: 1, 2, 7, 14, 29, 58, 203, 406
• Factor pairs: 406 = 1 x 406, 2 x 203, 7 x 58, or 14 x 29
• 406 has no square factors that allow its square root to be simplified. √406 ≈ 20.1494

### 405 and Level 3

405 times 2 equals 810, and 810 divided by 10 is 81. And guess what, 405 divided by 5 is also 81. This division trick is similar to a multiplication trick Mathemagicalsite blogged about.

All of 405’s factors are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 405 is a composite number.
• Prime factorization: 405 = 3 x 3 x 3 x 3 x 5, which can be written 405 = (3^4) x 5
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 405 has exactly 10 factors.
• Factors of 405: 1, 3, 5, 9, 15, 27, 45, 81, 135, 405
• Factor pairs: 405 = 1 x 405, 3 x 135, 5 x 81, 9 x 45, or 15 x 27
• Taking the factor pair with the largest square number factor, we get √405 = (√81)(√5) = 9√5 ≈ 20.1246

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.

### 404 and Level 2

It may seem like trivial trivia, but every factor of 404 is a palindrome. Scroll down past the puzzle to find out what all those factors are.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 404 is a composite number.
• Prime factorization: 404 = 2 x 2 x 101, which can be written 404 = (2^2) x 101
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 404 has exactly 6 factors.
• Factors of 404: 1, 2, 4, 101, 202, 404
• Factor pairs: 404 = 1 x 404, 2 x 202, or 4 x 101
• Taking the factor pair with the largest square number factor, we get √404 = (√4)(√101) = 2√101 ≈ 20.0998

### 403 and Level 1

Madam, I’m Adam! There is something palindromic about the factors of 403. What could they possibly be? Scroll down past the puzzle to see what they are.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

• 403 is a composite number.
• Prime factorization: 403 = 13 x 31
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 403 has exactly 4 factors.
• Factors of 403: 1, 13, 31, 403
• Factor pairs: 403 = 1 x 403 or 13 x 31
• 403 has no square factors that allow its square root to be simplified. √403 ≈ 20.0749