1549 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write.

Factors of 1549:

  • 1549 is a prime number.
  • Prime factorization: 1549 is prime.
  • 1549 has no exponents greater than 1 in its prime factorization, so √1549 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1549 has exactly 2 factors.
  • The factors of 1549 are outlined with their factor pair partners in the graphic below.

How do we know that 1549 is a prime number? If 1549 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1549. Since 1549 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1549 is a prime number.

More about the number 1549:

1549 is the sum of two squares:
35² + 18² = 1549

1549 is the hypotenuse of a Pythagorean triple:
901-1260-1549, calculated from 35² – 18², 2(35)(18), 35² + 18².

Here’s another way we know that 1549 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 18² = 1549 with 35 and 18 having no common prime factors, 1549 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1549. Since 1549 is not divisible by 5, 13, 17, 29, or 37, we know that 1549 is a prime number.

1543 Another Letter

Today’s Puzzle:

My goal was to make this next letter of my message using as few clues in the puzzle as possible. You now have enough clues to solve this puzzle and know my message to you.

Factors of 1543:

  • 1543 is a prime number.
  • Prime factorization: 1543 is prime.
  • 1543 has no exponents greater than 1 in its prime factorization, so √1543 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1543 has exactly 2 factors.
  • The factors of 1543 are outlined with their factor pair partners in the graphic below.

How do we know that 1543 is a prime number? If 1543 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1543. Since 1543 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1543 is a prime number.

One More Fact about the Number 1543:

1543 is the sum of consecutive numbers as well as the difference of those same consecutive numbers but squared:
771 + 772 = 1543;
772² – 771² = 1543.
(1543 has that property because it’s an odd number greater than 1.)

1537 First Letter of a Secret Message

Today’s Puzzle:

The next six puzzles will look like letters of the alphabet, and they will spell out a secret message for you. Will you figure out the message before I have published all of the letters? The difficulty levels of the puzzles will not be in any particular order.

Factors of 1537:

OEIS.org tells us that the largest prime factor of 1537 is 53. Pretty cool!

  • 1537 is a composite number.
  • Prime factorization: 1537 = 29 × 53.
  • 1537 has no exponents greater than 1 in its prime factorization, so √1537 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 1537 has exactly 4 factors.
  • The factors of 1537 are outlined with their factor pair partners in the graphic below.

More about the Number 1537:

1537 is the DIFFERENCE of two squares in two different ways:
769² – 768² = 1537, and
41² – 12² = 1537.

1537 is also the SUM of two squares in two different ways:
39² + 4² = 1537, and
31² + 24² = 1537.

1537 is the hypotenuse of FOUR Pythagorean triples:
312-1505-1537, calculated from 2(39)(4), 39² – 4², 39² + 4²,
385-1488-1537, calculated from 31² – 24², 2(31)(24), 31² + 24²,
812-1305-1537, which is 29 times (28-45-53), and
1060-1113-1537, which is (20-21-29) times 53.

 

 

1527 Not a Haunted House

Today’s Puzzle:

We see plenty of pumpkins on doorsteps this time of year, so I put a few in this puzzle. The puzzle looks a bit like a house, but certainly not a haunted house. Can you write the numbers from 1 to 10 in both the first column and the top row so that the given number clues are the products of those numbers?

ѼѼѼѼ 🎃🎃🎃🎃 ѽѽѽѽ  

Here’s the same puzzle without the added colorful embellishments:

Factors of 1527:

  • 1527 is a composite number.
  • Prime factorization: 1527 = 3 × 509
  • 1527 has no exponents greater than 1 in its prime factorization, so √1527 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 1527 has exactly 4 factors.
  • The factors of 1527 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1527:

1527 is the hypotenuse of a Pythagorean triple:
660-1377-1527, which is 3 times (220-459-509).

 

1514 and Level 2

Today’s Puzzle:

There is only one way to write the factors from 1 to 12 in both the first column and the top row so that this puzzle will behave like a multiplication table. The given clues will be the products of the factors you write. Can you find the way?

Factors of 1514:

  • 1514 is a composite number.
  • Prime factorization: 1514 = 2 × 757
  • 1514 has no exponents greater than 1 in its prime factorization, so √1514 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 1514 has exactly 4 factors.
  • The factors of 1514 are outlined with their factor pair partners in the graphic below.

More Facts about the number 1514:

1514 is the sum of two squares:
35² + 17² = 1514

1514 is the hypotenuse of a Pythagorean triple:
936-1190-1514 calculated from 35² – 17², 2(35)(17), 35² + 17²

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?

1490 and Level 2

Today’s Puzzle:

You can solve this puzzle if you know the basic multiplication and division facts. Just write the numbers from 1 to 12 in both the first column and the top row so the given clues and those factors create a valid multiplication table.

Factors of 1490:

  • 1490 is a composite number.
  • Prime factorization: 1490 = 2 × 5 × 149.
  • 1490 has no exponents greater than 1 in its prime factorization, so √1490 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 1490 has exactly 8 factors.
  • The factors of 1490 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1490:

1490 is the sum of two squares in two different ways:
31² + 23² = 1490.
37² + 11² = 1490.

1490 is the hypotenuse of FOUR Pythagorean triples:
432-1426-1490, calculated from 31² – 23², 2(31)(23), 31² + 23².
510-1400-1490, which is 10 times (51-140-149).
814-1248-1490, calculated from 2(37)(11), 37² – 11², 37² + 11².
894-1192-1490, which is (3-4-5) times 298.

1479 and Level 2

Today’s Puzzle:

Can you find the factors from 1 to 10 that will work with the clues in this puzzle to make a multiplication table?

Factors of 1479:

  • 1479 is a composite number.
  • Prime factorization: 1479 = 3 × 17 × 29
  • 1479 has no exponents greater than 1 in its prime factorization, so √1479 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 1479 has exactly 8 factors.
  • The factors of 1479 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1479:

1479 is the hypotenuse of FOUR Pythagorean triples:
396-1425-1479 which is 3 times (132-475-493)
465-1404-1479 which is 3 times (155-468-493)
696-1305-1479 which is (8-15-17) times 87
1020-1071-1479 which is (20-21-29) times 51

1469 and Level 2

Today’s Puzzle:

Where do the Factors 1-12 belong on this multiplication table puzzle?

Factors of 1469:

  • 1469 is a composite number.
  • Prime factorization: 1469 = 13 × 113
  • 1469 has no exponents greater than 1 in its prime factorization, so √1469 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 1469 has exactly 4 factors.
  • The factors of 1469 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1469:

1469 is the sum of two squares in two different ways:
38² + 5² = 1469
37² + 10² = 1469

1469 is the hypotenuse of FOUR Pythagorean triples:
195-1456-1469 which is 13 times (15-112-113)
380-1419-1469 calculated from 2(38)(5), 38² – 5², 38² + 5²
565-1356-1469 which is (5-12-13) times 113
740-1269-1469 calculated from 2(37)(10), 37² – 10², 37² + 10²

 

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.