1435 is the 23rd Friedman Number

41 ×35 = 1435. Since the same digits are used on both sides of the equal sign that makes 1435 the 23rd Friedman number.

Since it’s good for puzzles to have a puzzle number to distinguish them from each other, I’ve given this one the number 1435. Can you solve it?

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Here are a few more facts about the number 1435:

  • 1435 is a composite number.
  • Prime factorization: 1435 = 5 × 7 × 41
  • 1435 has no exponents greater than 1 in its prime factorization, so √1435 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1435 has exactly 8 factors.
  • The factors of 1435 are outlined with their factor pair partners in the graphic below.

1435 is also the hypotenuse of FOUR Pythagorean triples:
315-1400-1435 which is 35 times (9-40-41)
588-1309-1435 which is 7 times (84-187-205)
861-1148-1435 which is (3-4-5) times 287
931-1092-1435 which is 7 times (133-156-205)

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1434 and Level 3

If you know the common factors of 14 and 8, then you will have an excellent start to solve this puzzle. Once you place the factors of 18 and 8 in their proper places, just work down the puzzle row by row until all the factors from 1 to 10 are found.

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Now I’ll share some facts about the puzzle number, 1434:

  • 1434 is a composite number.
  • Prime factorization: 1434 = 2 × 3 × 239
  • 1434 has no exponents greater than 1 in its prime factorization, so √1434 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1434 has exactly 8 factors.
  • The factors of 1434 are outlined with their factor pair partners in the graphic below.

1434 is 234 in BASE 26 because
2(26²) + 3(26¹) + 4(26º) = 1434

1433 and Level 2

There is only one way to arrange the numbers from 1 to 10 in both the first column and the top row to make this puzzle function like a multiplication table. Can you find that one way?

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

The puzzle number is 1433. Here are some facts about that number:

  • 1433 is a prime number.
  • Prime factorization: 1433 is prime.
  • 1433 has no exponents greater than 1 in its prime factorization, so √1433 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1433 has exactly 2 factors.
  • The factors of 1433 are outlined with their factor pair partners in the graphic below.

How do we know that 1433 is a prime number? If 1433 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1433. Since 1433 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1433 is a prime number.

1433 is the sum of two squares:
37² + 8² = 1433

1433 is the hypotenuse of a Pythagorean triple:
592-1305-1433 calculated from 2(37)(8), 37² – 8², 37² + 8²

Here’s another way we know that 1433 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 8² = 1433 with 37 and 8 having no common prime factors, 1433 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1433. Since 1433 is not divisible by 5, 13, 17, 29, or 37, we know that 1433 is a prime number.

1432 and Level 1

If you can count by tens, then you can solve this Level 1 puzzle. I dare you to prove me wrong!

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Here is some information about the puzzle number, 1432:

  • 1432 is a composite number.
  • Prime factorization: 1432 = 2 × 2 × 2 × 179, which can be written 1432 = 2³ × 179
  • 1432 has at least one exponent greater than 1 in its prime factorization so √1432 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1432 = (√4)(√358) = 2√358
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1432 has exactly 8 factors.
  • The factors of 1432 are outlined with their factor pair partners in the graphic below.

1432 is 21212 in BASE 5 because
2(5⁴) + 1(5³) + 2(5²) + 1(5¹) + 2(5⁰) = 1432

 

1431 is a Triangular Number and a Hexagonal Number

If you only look at a list of triangular numbers or a list of hexagonal numbers, you might miss the relationship that figurate numbers have with each other.

1378 is the 52nd triangular number, and you can use it to find the 53rd triangular number (1431), the 53rd square number, the 53rd pentagonal number, and so forth.

351 is the 26th triangular number, and you can use it to find the 27th triangular number, the 27th square number, the 27th pentagonal number, the 27th hexagonal number (1431), and so forth.

See the relationship in the graphic below:

Should you get excited that 1431 is BOTH a triangular number and a hexagonal number? Not really. It turns out that every hexagonal number is also a triangular number. (But not every triangular number is a hexagonal number.)

Here are a few more facts about the number 1431:

  • 1431 is a composite number.
  • Prime factorization: 1431 = 3 × 3 × 3 × 53, which can be written 1431 = 3³ × 53
  • 1431 has at least one exponent greater than 1 in its prime factorization so √1431 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1431 = (√9)(√159) = 3√159
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1431 has exactly 8 factors.
  • The factors of 1431 are outlined with their factor pair partners in the graphic below.

1431 is the hypotenuse of a Pythagorean triple:
756-1215-1431 which is 27 times (28-45-53)

1430 Is a Catalan Number

1430 is the eighth Catalan number because it is equal to (2⋅8)!÷((8+1)!8!):
10 × 11 × 12 × 13 × 14 × 15 × 16 ÷( 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 ) = 1430.

For example, the vertices of a decagon can also be the vertices of eight triangles. Those eight triangles can be drawn in 1430 different ways. Here are a few of those ways:

Some of the 1430 ways are rotations and/or reflections of the ways illustrated above. Many of the 1430 ways are NOT represented in that graphic at all. It would be mind-boggling to draw all 1430 ways!

Here are some other facts about the number 1430:

  • 1430 is a composite number.
  • Prime factorization: 1430 = 2 × 5 × 11 × 13
  • 1430 has no exponents greater than 1 in its prime factorization, so √1430 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1430 has exactly 16 factors.
  • The factors of 1430 are outlined with their factor pair partners in the graphic below.

1430 is the hypotenuse of FOUR Pythagorean triples:
352-1386-1430 which is 22 times (16-63-65)
550-1320-1430 which is (5-12-13) times 110
726-1232-1430 which is 22 times (33-56-65)
858-1144-1430 which is (3-4-5) times 286

 

1429 Find the Factors Challenge

I wanted today’s puzzle to look like a big candy bar, but I don’t think I succeeded. I hope you will still think it is the best treat you got today! Good luck!

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you some facts about the number 1429:

  • 1429 is a prime number.
  • Prime factorization: 1429 is prime.
  • 1429 has no exponents greater than 1 in its prime factorization, so √1429 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1429 has exactly 2 factors.
  • The factors of 1429 are outlined with their factor pair partners in the graphic below.

How do we know that 1429 is a prime number? If 1429 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1429. Since 1429 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1429 is a prime number.

Stetson.edu informs us that 1429² = 2,042,041. That’s the smallest perfect square whose first three digits are repeated in order by the next three digits.

1429 is the sum of two squares:
30² + 23² = 1429

1429 is the hypotenuse of a primitive Pythagorean triple:
371-1380-1429 calculated from 30² – 23², 2(30)(23), 30² + 23²

Here’s another way we know that 1429 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 23² = 1429 with 30 and 23 having no common prime factors, 1429 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1429. Since 1429 is not divisible by 5, 13, 17, 29, or 37, we know that 1429 is a prime number.

 

1428 Factor Trees in Autumn

I recently decided that I wanted to make some factor trees in various fall colors. 1428 has plenty of factors so it has MANY different factor trees. Here are just eleven of them, each initially factored by a different factor pair.

Here are some more facts about the number 1428:

  • 1428 is a composite number.
  • Prime factorization: 1428 = 2 × 2 × 3 × 7 × 17, which can be written 1428 = 2² × 3 × 7 × 17
  • 1428 has at least one exponent greater than 1 in its prime factorization so √1428 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1428 = (√4)(√357) = 2√357
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1428 has exactly 24 factors.
  • The factors of 1428 are outlined with their factor pair partners in the graphic below.

1428 is the hypotenuse of a Pythagorean triple:
672-1260-1428 which is (8-15-17) times 84.

1427 Mysterious Cat

This mysterious Halloween cat shares twelve clues that can help you solve its puzzle. Each clue is the products of two factors from 1 to 12 multiplied together. Will you be able to solve its mystery?

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you a little bit about the puzzle number, 1427:

  • 1427 is a prime number.
  • Prime factorization: 1427 is prime.
  • 1427 has no exponents greater than 1 in its prime factorization, so √1427 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1427 has exactly 2 factors.
  • The factors of 1427 are outlined with their factor pair partners in the graphic below.

How do we know that 1427 is a prime number? If 1427 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1427. Since 1427 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1427 is a prime number.

1427 is 272 in BASE 25 because 2(25²) + 7(25) + 2(1) = 1427

1426 is a Pentagonal Number

I remembered that 1426 is a pentagonal number, but I didn’t have the formula for pentagonal numbers memorized. While I was waiting to do something else, I tried to come up with the formula myself.

The formula I derived isn’t what you usually see, but I rather like it! This formula can be extended to any figurate number as the chart below shows.

I love that the distance between consecutive figurate numbers on the chart is 465 which is the 30th triangular number!

Now if you asked me how many little dots does the 31st 1000-gonal number have, I also would be able to tell you that it has 998(465) + 31 = 464101 dots!

But today’s post is asking for the 31st pentagonal number. Here is what that looks like:

Can you see that it is three times the 30th triangular number plus 31?

Here are some more facts about the number 1426:

  • 1426 is a composite number.
  • Prime factorization: 1426 = 2 × 23 × 31
  • 1426 has no exponents greater than 1 in its prime factorization, so √1426 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1426 has exactly 8 factors.
  • The factors of 1426 are outlined with their factor pair partners in the graphic below.

Since 1426 = 2(31)(23) and 2(713)(1), it is a leg in two Pythagorean triples:
432-1426-1490 calculated from 31² – 23², 2(31)(23), 31² + 23²
1426-508368-508370 calculated from 2(713)(1), 713² – 1², 713² + 1²