A Multiplication Based Logic Puzzle

Archive for January, 2015

375 and Level 6

375 is a composite number because its last digit is 5. Scroll down below the puzzle to see its factoring information.

375 Puzzle

  • 375 is a composite number.
  • Prime factorization: 375 = 3 x 5 x 5 x 5, which can be written 375 = 3 x (5^3)
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 x 4 = 8. Therefore 375 has exactly 8 factors.
  • Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
  • Factor pairs: 375 = 1 x 375, 3 x 125, 5 x 75, or 15 x 25
  • Taking the factor pair with the largest square number factor, we get √375 = (√15)(√25) = 5√15 ≈ 19.365

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

375 Logic

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374 and Level 5

374 is even so it is a composite number. Its factoring information is below the puzzle.

374 Puzzle

  • 374 is a composite number.
  • Prime factorization: 374 = 2 x 11 x 17
  • 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 374 has exactly 8 factors.
  • Factors of 374: 1, 2, 11, 17, 22, 34, 187, 374
  • Factor pairs: 374 = 1 x 374, 2 x 187, 11 x 34, or 17 x 22
  • 374 has no square factors that allow its square root to be simplified. √374 ≈ 19.339

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

374 Logic

373 Prime Numbers and Pythagorean Triples

  • 373 is a prime number.
  • Prime factorization: 373 is prime and cannot be factored.
  • The exponent of prime number 373 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 373 has exactly 2 factors.
  • Factors of 373: 1, 373
  • Factor pairs: 373 = 1 x 373
  • 373 has no square factors that allow its square root to be simplified. √373 ≈ 19.313

How do we know that 373 is a prime number? If 373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √373 ≈ 19.313. Since 373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 373 is a prime number.

  • 373 is the short leg of only one Pythagorean triple, the primitive 373, 69564, 69565
  • Prime numbers are never the longer leg, but
  • 373 is the hypotenuse of exactly one Pythagorean triple: 252, 275, 373

Here is the Odd Pythagorean triple sequence I’ve blogged about this week with the prime numbers highlighted in yellow:

Odd Primitive Primes

There are 24 odd prime numbers less than 100. The odd numbers less than 100 in this sequence produce 21 prime numbers as their hypotenuses! I think that is amazing especially since 40% of the time the hypotenuse turns out to be a composite number whose last digit is five! Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 3 or 7, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 5, the last digit of the hypotenuse ends in 3.
  • When the last digit of the short leg is 1 or 9, the last digit of the hypotenuse ends in 1.

There are only 18 prime hypotenuses when we use about the same number of triples from this Even Primitive Triple Sequence.

Even Primitive Primes

18 primes out of 49 numbers listed is slightly less impressive than 21 primes out of 48 total numbers, but again 40% of the hypotenuses end with five and have no choice but to be composite numbers.  Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 4 or 6, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 0, the last digit of the hypotenuse ends in 1.
  • When the last digit of the short leg is 2 or 8, the last digit of the hypotenuse ends in 7.

I had no idea that Pythagorean triples would produce so much trivia!

372 and Level 4

372 is even and therefore a composite number. Scroll down for its factoring information.

372 Puzzle

  • 372 is a composite number.
  • Prime factorization: 372 = 2 x 2 x 3 x 31, which can be written 372 = (2^2) x 3 x 31
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 372 has exactly 12 factors.
  • Factors of 372: 1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, 372
  • Factor pairs: 372 = 1 x 372, 2 x 186, 3 x 124, 4 x 93, 6 x 62, or 12 x 31
  • Taking the factor pair with the largest square number factor, we get √372 = (√4)(√93) = 2√93 ≈ 19.287

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

372 Logic

371 What Pythagorean Triple Comes Next?

2^371 (mod 371) = 340, not 2. Therefore, 371 is definitely a composite number. Scroll down to see its factoring information.

What Pythagorean Triple Comes Next

If you can figure out the simple pattern that these odd Pythagorean triples make, you can predict the next one in the sequence FOREVER by squaring only one number, and without ever taking a single square root! Look at the blue and red squares in the graphic above.

  • 4 + 5 = 9 = 3^2, and 3, 4, 5 make a Pythagorean triple.
  • 12 + 13 = 25 = 5^2, and 5, 12, 13 is also a Pythagorean triple.
  • 24 + 25 = 49 = 7^2, and 7, 24, 25 is another Pythagorean triple, and
  • 40 + 41 = 81 = 9^2, and 9, 40, 41 is yet another Pythagorean triple!

What Pythagorean triple comes next? The answer is found by taking the next odd number, 11, squaring it to get 121 and dividing the 121 by 2. We round down to 60 for the 2nd number and round up to 61 for the 3rd number. Thus the next Pythagorean triple would be 11, 60, 61.

No matter how many times we repeat this pattern, we always get primitive Pythagorean triples. We can put all the triples in order to make a sequence that I call the Odd Primitive Pythagorean Triple Sequence:

The Odd Primitive Pythagorean Triple Sequence

Every odd number is the short leg of at least one primitive Pythagorean triple! Here’s how I came to realize this amazing fact:

Last week I was thinking about the difference of two squares applied to integers in general

Difference of Two Squares

as I looked at this multiplication table:

The Multiplication Table

Everyone knows that the numbers in the boxes outlined in red are perfect squares, but most people do not realize that the numbers inside EVERY other colored square on this multiplication table can be expressed as the difference of two squares. The larger of those two squares will be the perfect square that is the same color.

For example, look at 5 x 11 or 55. If we count as we follow the light blue diagonal stripe from 55 to 64, we count 3 squares. That means that 55 = 64 – 3^2.

As I looked at the multiplication table I realized that some even numbered squares are colored and some are not, but EVERY odd number square is in color! That means that EVERY odd number on the table can be expressed as the difference of two squares in at least one way.

The light blue 9 from 1 x 9 is four squares away from the 25 that is outlined in red. 9 = 25 – 4^2. Since all those numbers are perfect squares, we can write 3^2 = 5^2 – 4^2 or the equivalent in Pythagorean Theorem form   3^2 + 4^2 = 5^2.

When an odd number is squared, the resulting perfect square is ALWAYS an odd number. Since all odd numbers can be written as the difference of two squares, The square of all odd numbers can be written as the difference of two squares. In other words, every odd number is a leg in a Pythagorean triple!

Let’s look at the factoring information for 371 and use it to find some Pythagorean triples.

  • 371 is a composite number.
  • Prime factorization: 371 = 7 x 53
  • 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 371 has exactly 4 factors.
  • Factors of 371: 1, 7, 53, 371
  • Factor pairs: 371 = 1 x 371 or 7 x 53
  • 371 has no square factors that allow its square root to be simplified. √371 ≈ 19.261

371 is odd and has two odd prime factors. Here are some Pythagorean triples involving 371 as the short leg:

  • 371, 68820, 68821 (This triple is a primitive from the odd primitive sequence.)
  • 371, 1380, 1429 (A primitive, not from the odd primitive sequence. I’m also making a note that √(1380 + 1429) = 53 and √(1429 – 1380) = 7.)
  • 7, 24, 25 multiplied by 53 becomes 371, 1272, 1325
  • 53, 1404, 1405 multiplied by 7 becomes 371, 9828, 9835

371 is also the hypotenuse of a Pythagorean triple: 196, 315, 371 which is primitive 28, 45, 53 multiplied by 7.

370 and Level 3

370 is a composite number because it is a multiple of ten. Scroll down to see its factors.

370 Puzzle

  • 370 is a composite number.
  • Prime factorization: 370 = 2 x 5 x 37
  • 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 370 has exactly 8 factors.
  • Factors of 370: 1, 2, 5, 10, 37, 74, 185, 370
  • Factor pairs: 370 = 1 x 370, 2 x 185, 5 x 74, or 10 x 37
  • 370 has no square factors that allow its square root to be simplified. √370 ≈ 19.235

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

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.

370 Factors

369 and Level 2

3, 6, & 9 are each divisible by 3, so 369 is composite and a multiple of 3. Scroll down past the puzzle to see its factors.

369 Puzzle

  • 369 is a composite number.
  • Prime factorization: 369 = 3 x 3 x 41, which can be written 369 = (3^2) x 41
  • 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 369 has exactly 6 factors.
  • Factors of 369: 1, 3, 9, 41, 123, 369
  • Factor pairs: 369 = 1 x 369, 3 x 123, or 9 x 41
  • Taking the factor pair with the largest square number factor, we get √369 = (√9)(√41) = 3√41 ≈ 19.209

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

369 Factors

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