# 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.

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

• 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

# 374 and Level 5

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

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

• 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

# 373 Prime Numbers and Pythagorean Triples

• 373 is a prime number.
• Prime factorization: 373 is prime.
• 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:

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.

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.

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

• 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

# 371 What Pythagorean Triple Comes Next?

Could 371 be a prime number? Let’s do a quick prime number test on it.  2^371 (mod 371) = 340, not 2. Therefore, 371 is definitely a composite number. Scroll down to see its factoring information.

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 = 3², and 3, 4, 5 make a Pythagorean triple.
• 12 + 13 = 25 = 5², and 5, 12, 13 is also a Pythagorean triple.
• 24 + 25 = 49 = 7², and 7, 24, 25 is another Pythagorean triple, and
• 40 + 41 = 81 = 9², 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:

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

as I looked at this 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 – 3².

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 – 4². Since all those numbers are perfect squares, we can write 3² = 5² – 4² or the equivalent in Pythagorean Theorem form   3² + 4² = 5².

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.

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

• 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

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.

# 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.

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

• 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

# 368 and a Few Sequences

368 is a composite number because it is even. Scroll down for its factoring information.

Sequences can be illustrated in pictures, but they are most often simply a list of numbers such as 23, 46, 69, 92, 115, . . . . .

Those numbers are the multiples of 23 listed in order. If we continued writing the numbers in that sequence, we would see that 368 is the 16th number listed.

We also know that 368 is an element of many other sequences even though it isn’t one of the typed numbers:

• The sequence of all counting numbers: 1, 2, 3, 4, 5, . . . .
• The sequence of all even counting numbers: 2, 4, 6, 8, 10, . . . .
• The sequence of numbers that are multiples of four: 4, 8, 12, 16, 20, . . . . (This is the yellow sequence in the graphic below.)
• The sequence of numbers that are one less than the multiples of three: 2, 5, 8, 11, 14, . . . .
• The sequence of numbers that are one less than the multiples of nine: 8, 17, 26, 35, 44, . . . .

386 is NOT an element of any of the following sequences that involve even squared numbers:

• 3, 15, 35, 63, 99, . . . . (The blue sequence in the graphic above)
• 4, 16, 36, 64, 100, . . . .
• 5, 17, 37, 65, 101, . . . . (The red sequence in the graphic above)

The yellow, blue, and red sequences above can be made into an array that contains only Primitive Pythagorean Triples!

Except for the 3, 4, 5 triangle that is listed first, every multiple of 4 is the short leg in a primitive Pythagorean triple! The other leg and the hypotenuse are based on a square with sides that are 1/2 the length of the shorter leg. Since the shorter leg is a multiple of four, that square length will always be an even number. Also note: 368 is divisible by 4 so 368, 33855, 33857 is a primitive that would be included in this sequence array.

The sequence contains only primitives, but it does NOT contain EVERY primitive whose short leg is a multiple of four. For example Pythagorean triple, 20, 21, 29, is not in that sequence array.

Let’s look at all the factoring information for 368:

• 368 is a composite number.
• Prime factorization: 368 = 2 x 2 x 2 x 2 x 23, which can be written 368 = (2^4) x 23
• 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 368 has exactly 10 factors.
• Factors of 368: 1, 2, 4, 8, 16, 23, 46, 92, 184, 368
• Factor pairs: 368 = 1 x 368, 2 x 184, 4 x 92, 8 x 46, or 16 x 23
• Taking the factor pair with the largest square number factor, we get √368 = (√16)(√23) = 4√23 ≈ 19.183

We can make a sequence array from the Pythagorean triple primitive 3, 4, 5 and its non-primitive multiples (which includes 276, 368, 460):

We can find other Pythagorean triples that contain the number 368. Notice that 368 has several factors that are multiples of 4. They are 4, 8, 16, 92, and 184. Each one of them has its own primitive triple which can be multiplied by its factor pair partner to produce non-primitive triples that include 368 as well.

• 3, 4, 5 multiplied by 92 is 276, 368, 460
• 8, 15, 17 multiplied by 46 is 368, 690, 782
• 16, 63, 65 multiplied by 23 is 368, 1449, 1495
• 92, 2115, 2117 multiplied by 4 is 368, 8460, 8468
• 184, 8463, 8465 multiplied by 2 is 368, 16926, 16930

In future posts I’ll write about how to find other Pythagorean triples like

• 23, 264, 265, a primitive Pythagorean triple with an odd short leg. This primitive becomes 368, 4224, 4240 when multiplied by 16 (23’s factor pair partner).
• 368, 465, 593, another primitive Pythagorean triple not included in the colored sequence array above and
• 184, 513, 545, which when multiplied by 2 becomes 368, 1026, 1090.

Pythagorean triples aren’t just a bunch of “SQUARE” numbers. They are simply counting numbers that satisfy the equation a² + b² = c², and they are really COOL!

# 367 and Level 1

2^367 (mod 367) = 2, so 367 is VERY LIKELY a prime number. Scroll down below the puzzle to know for sure.

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

• 367 is a prime number.
• Prime factorization: 367 is prime.
• The exponent of prime number 367 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 367 has exactly 2 factors.
• Factors of 367: 1, 367
• Factor pairs: 367 = 1 x 367
• 367 has no square factors that allow its square root to be simplified. √367 ≈ 19.157

How do we know that 367 is a prime number? If 367 were not a prime number, then it would be divisible by at least one prime number less than or equal to √367 ≈ 19.157. Since 367 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 367 is a prime number.

# 366 and a Quote from Euclid

The laws of nature are but the mathematical thoughts of God.  ~Euclid

Euclid was a Greek mathematician who was born about 300 BC. I realize when he wrote these words (in ancient Greek), they didn’t mean the exact same thing to him that they mean to me. Nevertheless, I like the quote very much, and find infinite beauty in its message. The laws of nature are but the mathematical thoughts of God.  ~Euclid

Since writing this post, I have found some other lovely representations of this quote on twitter:

It is well worth pushing the button to see this next gif in action:

Here are some mathematical thoughts about the number 366:

• 366 is a composite number.
• Prime factorization: 366 = 2 x 3 x 61
• 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 366 has exactly 8 factors.
• Factors of 366: 1, 2, 3, 6, 61, 122, 183, 366
• Factor pairs: 366 = 1 x 366, 2 x 183, 3 x 122, or 6 x 61
• 366 has no square factors that allow its square root to be simplified. √366 ≈ 19.131

66² + 360² = 366² How do I know? Because 11² + 60² = 61² so (6·11)² + (6·60)² = (6·61)².

366 is a palindrome in a couple of bases:

• 303 in BASE 11 because 3(11²) + 0(11¹) + 3(11º) = 366
• 222 in BASE 13 because 2(13²) + 2(13¹) + 2(13º) = 366

These are just a few of the things known by the Greatest Mathematician of All! He completely understands EVERYTHING from infinitesimal to infinity. By His grace I understand a tiny bit of what He knows.