1504 and Level 3

Today’s Puzzle:

Since this is a level 3 puzzle the clues are given in a logical order from top to bottom. Write the factors 1 to 10 in the first column and again in the top row.

Usually, you only have to consider the previous clues when finding the factors in a level 3 puzzle, but when you consider if 4 = 2 × 2 or 1 × 4, you will also have to look at a clue below it. You can do this!

Factors of 1504:

  • 1504 is a composite number.
  • Prime factorization: 1504 = 2 × 2 × 2 × 2 × 2 × 47, which can be written 1504 = 2⁵ × 47
  • 1504 has at least one exponent greater than 1 in its prime factorization so √1504 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1504 = (√16)(√94) = 4√94
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1504 has exactly 12 factors.
  • The factors of 1504 are outlined with their factor pair partners in the graphic below.

More about the Number 1504:

1504 is the difference of two squares in four different ways:
377² – 375² = 1504
190² – 186² = 1504
98² – 90² = 1504
55² – 39² = 1504

Why is 1503 a Friedman Number?

Friedman Puzzle:

Can you find an expression equaling 1503 that uses 1, 5, 0, and 3 each exactly once, but in any order, and some combination of  +, -, ×, or ÷? For this particular Friedman puzzle, none of those digits are exponents. If you can solve this Friedman puzzle, you will know why 1503 is the 24th Friedman number. You can find the solution hidden someplace in this post. (By the way, another permutation of those digits, 1530, will be the 25th Friedman number!)

Find the Factors Puzzle:

There are 14 clues in this level 2 puzzle. Use those clues and logic to place the factors 1 to 10 in both the first column and the top row. That’s how you start to turn this puzzle into a multiplication table!

Factors of 1503:

  • 1503 is a composite number.
  • Prime factorization: 1503 = 3 × 3 × 167, which can be written 1503 = 3² × 167
  • 1503 has at least one exponent greater than 1 in its prime factorization so √1503 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1503 = (√9)(√167) = 3√167
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1503 has exactly 6 factors.
  • The factors of 1503 are outlined with their factor pair partners in the graphic below.

Did you see the solution to the Friedman puzzle in that factor pair chart?

1502 and Level 1

Today’s Puzzle:

This level 1 puzzle has products in one of the rows and in one of the columns. Can you use those products to figure out where the factors 1 to 10 belong in this multiplication table puzzle?

Factors of 1502:

  • 1502 is a composite number.
  • Prime factorization: 1502 = 2 × 751
  • 1502 has no exponents greater than 1 in its prime factorization, so √1502 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1502 has exactly 4 factors.
  • The factors of 1502 are outlined with their factor pair partners in the graphic below.

Pythagorean Triple with 1502:

1502 is not the sum or the difference of two squares, but 1502 = 2(751)(1), so it is part of a Pythagorean triple:
1502-564000-564002, calculated from 2(751)(1), 751² – 1², 751² – 1².

1501 is a Centered Pentagonal Number

1501 Tiny Dot Puzzle

A long time ago before calculators or computers existed, someone determined that 1501 tiny dots could be formed into a pentagon. That was a remarkable puzzle to complete! What kind of pentagon will 1501 tiny dots make?

Since 5(24)(25)/2 + 1 = 1501, it is the 25th centered pentagonal number.  Notice in the graphic below that those 1501 tiny dots can also be divided into 5 equally-sized triangles with just the center dot leftover.

Factors of 1501:

  • 1501 is a composite number.
  • Prime factorization: 1501 = 19 × 79
  • 1501 has no exponents greater than 1 in its prime factorization, so √1501 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1501 has exactly 4 factors.
  • The factors of 1501 are outlined with their factor pair partners in the graphic below.

Another fact about the number 1501:

1501 is the difference of two squares in two different ways:
751² – 750² = 1501,
49² – 30² = 1501.

Celebrating 1500 with a Horse Race and Much More!

Pick Your Pony!

Writing 1500 posts is quite a milestone. I’ll begin the celebration with an exciting horse race! Let me explain:

Each prime number has exactly 2 factors. Every composite number between 1401 and 1500 has somewhere between 4 and 36 factors. Which quantity of factors do you think will appear most often for these numbers? Pick that amount as your pony and see how far it gets in this horse race!
1401 to 1500 Horse Race

make science GIFs like this at MakeaGif

Did the race results surprise you? They surprised me!

Prime Factorization for numbers from 1401 to 1500:

Here’s a chart showing the prime factorization of all those numbers and the amount of factors each number has. Numbers in pink have exponents in their prime factorization so their square roots can be simplified:

Today’s Puzzles:

Let’s continue the celebration with a puzzle: 1500 has 12 different factor pairs. One of those pairs adds up to 85 and one of them subtracts to give 85. Can you find those factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1500 in the graphic after the puzzle, but the second puzzle is really just the first puzzle in disguise. So try solving that easier puzzle first.

Factors of 1500:

  • 1500 is a composite number.
  • Prime factorization: 1500 = 2 × 2 × 3 × 5 × 5 × 5, which can be written 1500 = 2² × 3 × 5³
  • 1500 has at least one exponent greater than 1 in its prime factorization so √1500 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1500 = (√100)(√15) = 10√15
  • The exponents in the prime factorization are 2, 1, and 3. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(3 + 1) = 3 × 2 × 4 = 24. Therefore 1500 has exactly 24 factors.
  • The factors of 1500 are outlined with their factor pair partners in the graphic below.

A Forest of Factor Trees:

Take a few minutes to hike in this forest featuring a few of the MANY possible factor trees for 1500. Celebrate each tree’s uniqueness!

Other Facts to Celebrate about 1500:

Oeis.org tells us that  (5+1) × (5+5) × (5+0) × (5+0) = 1500.

1500 is the hypotenuse of THREE Pythagorean triples:
420-1440-1500, which is (7-24-25) times 60,
528-1404-1500, which is 12 times (44-117-225),
900-1200-1500, which is (3-4-5) times 300.

1499 Challenge Puzzle

Today’s Puzzle:

Use the 19 clues, logic, and the multiplication facts from a 10 × 10 multiplication table to find the unique solution of this Find the Factors Challenge puzzle. Good luck!

Factors of 1499:

  • 1499 is a prime number.
  • Prime factorization: 1499 is prime.
  • 1499 has no exponents greater than 1 in its prime factorization, so √1499 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1499 has exactly 2 factors.
  • The factors of 1499 are outlined with their factor pair partners in the graphic below.

How do we know that 1499 is a prime number? If 1499 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1499. Since 1499 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1499 is a prime number.

More Facts about the Number 1499:

1499 is the difference of two consecutive squares:
750² – 749² = 1499

oeis.org reminds us that 149, 199, and 499 are also prime numbers, so taking away one digit from 1499 always leaves a prime number.

 

 

1498 Another Mystery

Today’s Puzzle:

The number 48 appears four times in a 12 × 12 multiplication table, and all four 48’s appear in this puzzle! Where will you put its factors, 6, 8, 4, 12? Use logic to figure out where all the numbers from 1 to 12 need to go to make this puzzle turn into a multiplication table.

Factors of 1498:

  • 1498 is a composite number.
  • Prime factorization: 1498 = 2 × 7 × 107.
  • 1498 has no exponents greater than 1 in its prime factorization, so √1498 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 1498 has exactly 8 factors.
  • The factors of 1498 are outlined with their factor pair partners in the graphic below.

Pythagorean Triples with 1498:

1498 is 2(107)(7) so we can calculate a cool Pythagorean triple from
2(107)(7), 107² – 7², 107² + 7² to get 1498-11400-11498.

Also from 2(749)(1), 749² – 1², 749² + 1², we get 1498-561000-561002.

 

1497 Mystery

Today’s Puzzle:

Knowing where to place some of the numbers from 1 to 12 in this puzzle shouldn’t be too difficult, but placing ALL of the numbers will be like solving a mystery.

Factors of 1497:

  • 1497 is a composite number.
  • Prime factorization: 1497 = 3 × 499
  • 1497 has no exponents greater than 1 in its prime factorization, so √1497 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1497 has exactly 4 factors.
  • The factors of 1497 are outlined with their factor pair partners in the graphic below.

More about the Number 1497:

1497 is the difference of two squares in two different ways:
749² – 748² = 1497
251² – 248² = 1497

1496 is a Square Pyramidal Number

Visualize 1496 Blocks:

I hoped to make a graphic illustrating that 1496 is the 16th square pyramidal number. I am thrilled that I succeeded!

Factors of 1496:

Knowing the multiplication table and some divisibility tricks helped me find some of 1496’s factors:
96 is 8 × 12, and 4 is even, so 1496 is divisible by 8.
1 – 4 + 9 – 6 = 0, so 1496 is divisible by 11.

  • 1496 is a composite number.
  • Prime factorization: 1496 = 2 × 2 × 2 × 11 × 17, which can be written 1496 = 2³ × 11 × 17
  • 1496 has at least one exponent greater than 1 in its prime factorization so √1496 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1496 = (√4)(√374) = 2√374
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1496 has exactly 16 factors.
  • The factors of 1496 are outlined with their factor pair partners in the graphic below.

Another fact about the number 1496:

1496 is the hypotenuse of a Pythagorean triple:
704-1320-1496, which is (8-15-17) times 88.

1495 and Level 6

Today’s Puzzle:

Hint: The only way 12 can be put in the first column of this puzzle is to let one of the 60’s use it. We don’t have to know which 60 that is, to know that a 5 will go above that 60 in the top row. Knowing that, where does 5 have to go in the first column? That’s the logic needed to get started on this puzzle:

Factors of 1495:

  • 1495 is a composite number.
  • Prime factorization: 1495 = 5 × 13 × 23.
  • 1495 has no exponents greater than 1 in its prime factorization, so √1495 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 1495 has exactly 8 factors.
  • The factors of 1495 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1495:

1495 is the hypotenuse of FOUR Pythagorean triples:
368-1449-1495, which is 23 times (16-63-65)
575-1380-1495, which is (5-12-13) times 115,
759-1288-1495, which is 23 times (33-56-65), and
897-1196-1495, which is (3-4-5) times 299.