1599 and Level 5

Today’s Puzzle:

Write the numbers 1 to 10 in both the first column and the top row so that this level 5 puzzle will function like a multiplication table. Use logic with every step.

Factors of 1599:

  • 1599 is a composite number.
  • Prime factorization: 1599 = 3 × 13 × 41.
  • 1599 has no exponents greater than 1 in its prime factorization, so √1599 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 1599 has exactly 8 factors.
  • The factors of 1599 are outlined with their factor pair partners in the graphic below.

More about the Number 1599:

1599 is the hypotenuse of FOUR Pythagorean triples:
276-1575-1599, which is 3 times (92-525-533),
351-1560-1599, which is 39 times (9-40-41),
615-1476-1599, which is (5-12-13) times 123, and
924-1305-1599, which is 3 times (308-435-533).

1599 is the difference of two squares four different ways:
800² – 799² = 1599,
268² – 265² = 1599,
68² – 55² = 1599, and
40² – 1² = 1599.
Yes, we are just one number away from a perfect square!

1590 A Single Rose

Today’s Puzzle:

A single rose can be an elegant expression of affection. This single rose is a level 5 puzzle. Can you find its factors?

Here’s the same puzzle without any added color:

Two Factor Trees for 1590:

There are several possible factor trees for 1590. Here are two of them.

Factors of 1590:

  • 1590 is a composite number.
  • Prime factorization: 1590 = 2 × 3 × 5 × 53
  • 1590 has no exponents greater than 1 in its prime factorization, so √1590 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 1590 has exactly 16 factors.
  • The factors of 1590 are outlined with their factor pair partners in the graphic below.

More about the Number 1590:

1590 is the hypotenuse of FOUR Pythagorean triples:
138-1584-1590, which is 6 times (23-264-265),
576-1482-1590, which is 6 times (96-247-265)
840-1350-1590, which is 30 times (28-45-53),
954-1272-1590, which is (3-4-5) times 318.

 

1581 One Teardrop among Millions

Today’s Puzzle:

Earlier this week the United States surpassed 400,000 deaths from the novel coronavirus. Over 2,000,000 people have died from the virus worldwide. Most of these people died in isolation away from loved ones. Most of the millions of tears shed have also been in isolation. New and more contagious strains of the virus have arisen in various parts of the world threatening to make the death toll worse and the number of tears to grow exponentially. Each of us can and must do our part to flatten the curve and thus restrict the number of deaths and the number of tears.

Factors of 1581:

  • 1581 is a composite number.
  • Prime factorization: 1581 = 3 × 17 × 31.
  • 1581 has no exponents greater than 1 in its prime factorization, so √1581 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 1581 has exactly 8 factors.
  • The factors of 1581 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1581:

1581 is the hypotenuse of a Pythagorean triple:
744-1395-1581, which is (8-15-17) times 93.

1563 The Holly Wreath

Today’s puzzle:

A holly wreath is yet another symbol that connects Christmas with Easter. It symbolizes eternity in its color and shape. It bears white flowers, red berries, and thorns reminding us of purity, blood, and a crown of thorns.

You might find some of the clues in this level 5 puzzle to be like thorns, but don’t give up. Use logic and perseverance and you will be able to find its unique solution.

Here’s the same puzzle without all the added color:

Factors of 1563:

  • 1563 is a composite number.
  • Prime factorization: 1563 = 3 × 521.
  • 1563 has no exponents greater than 1 in its prime factorization, so √1563 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 1563 has exactly 4 factors.
  • The factors of 1563 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1563:

1563 is the hypotenuse of a Pythagorean triple:
837-1320-1563, which is 3 times (279-440-521).

1552 Look for Clues around the Corner

Today’s Puzzle:

This puzzle has four sets of clues that turn the corner. You will need to look around those corners to solve it. Use logic and have fun!

Factors of 1552:

  • 1552 is a composite number.
  • Prime factorization: 1552 = 2 × 2 × 2 × 2 × 97, which can be written 1552 = 2⁴ × 97
  • 1552 has at least one exponent greater than 1 in its prime factorization so √1552 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1552 = (√16)(√97) = 4√97
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1552 has exactly 10 factors.
  • The factors of 1552 are outlined with their factor pair partners in the graphic below.

More about the Number 1552:

1552 is the sum of two squares:
36² + 16²  = 1552

1552 is the hypotenuse of a Pythagorean triple:
1040-1152-1552, calculated from 36² – 16², 2(36)(16), 36² + 16².
It is also 16 times (65-72-97).

OEIS.org looked around the corner at the two numbers preceding 1552 to find something special about that number: The sum of its prime factors equals the sum of the prime factors of those previous two numbers! That’s a cool enough fact that I decided to make this graphic:

1530 Jack-o’-lantern

Today’s Puzzle:

Here’s a Jack-O’-Lantern Puzzle for you to enjoy. It’s a Level 5 puzzle so it might be more of a trick than a treat. Remember to use logic every step of the way instead of guessing and checking.

Here’s the same puzzle without any added color:

Factors of 1530:

15 is half of 30, so 1530 is divisible by 6 just like all these numbers are divisible by 6: 12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, and so forth.

  • 1530 is a composite number.
  • Prime factorization: 1530 = 2 × 3 × 3 × 5 × 17, which can be written 1530 = 2 × 3² × 5 × 17
  • 1530 has at least one exponent greater than 1 in its prime factorization so √1530 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1530 = (√9)(√170) = 3√170
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1530 has exactly 24 factors.
  • The factors of 1530 are outlined with their factor pair partners in the graphic below.

More about the Number 1530:

51 × 30 = 1530. Did you notice that the same digits appear on both sides of the equal sign and only +, -, ×, ÷, (), or exponents were used to make a true statement? 1530 is only the 25th number that can make that claim, so we call it the 25th Friedman number.

There are MANY possible factor trees for 1530, but let’s celebrate that it is also a Friedman number with this one:

1530 is the hypotenuse of FOUR Pythagorean triples:
234-1512-1530, which is 18 times (13-84-85),
648-1386-1530, which is 18 times (36-77-85),
720-1350-1530, which is (8-15-17) times 90, and
918-1224-1530, which is (3-4-5) times 306.

 

 

1517 and Level 5

Today’s Puzzle:

Which common factor of 72 and 36 is needed to solve this puzzle? Is it 6, 9, or 12? There is an easier place to begin this level 5 puzzle. Don’t guess and check. Use logic to know which factors you should use.  You can figure it out!

Factors of 1517:

  • 1517 is a composite number.
  • Prime factorization: 1517 = 37 × 41
  • 1517 has no exponents greater than 1 in its prime factorization, so √1517 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 1517 has exactly 4 factors.
  • The factors of 1517 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1517:

1517 is the difference of two squares in two different ways:
759² – 758² = 1517,
39² – 2² = 1517.

1517 is also the sum of two squares in two different ways:
34² + 19² = 1517,
29² + 26² = 1517.

1517 is the hypotenuse of FOUR Pythagorean triples:
165-1508-1517, calculated from 29² – 26², 2(29)(26), 29² + 26²,
333-1480-1517, which is 37 times (9-40-41),
492-1435-1517, which is (12-35-37) times 41,
795-1292-1517, calculated from 34² – 19², 2(34)(19), 34² + 19².

1506 and Level 5

Today’s Puzzle:

Can you find the factors 1 to 10 in a logical order so that the given clues are the products of those factors? Don’t let any of the clues trick you!

Factors of 1506:

  • 1506 is a composite number.
  • Prime factorization: 1506 = 2 × 3 × 251.
  • 1506 has no exponents greater than 1 in its prime factorization, so √1506 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 1506 has exactly 8 factors.
  • The factors of 1506 are outlined with their factor pair partners in the graphic below.

Pythagorean triples with 1506:

1506 is not the sum or the difference of two squares but it is still part of two Pythagorean triples:
1506-567008-567010, calculated from 2(753)(1), 753² – 1², 753² + 1², and
1506-62992-63010, calculated from 2(251)(3), 251² – 1², 251² + 1².

1494 and Level 5

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of those factors. Be sure to use logic to find the factors! Guessing and checking will only frustrate you.

Factors of 1494:

  • 1494 is a composite number.
  • Prime factorization: 1494 = 2 × 3 × 3 × 83, which can be written 1494 = 2 × 3² × 83
  • 1494 has at least one exponent greater than 1 in its prime factorization so √1494 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1494 = (√9)(√166) = 3√166
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1494 has exactly 12 factors.
  • The factors of 1494 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1494:

1494 is not the sum of or the difference of two squares, but it is still a part of three Pythagorean triples because of these three ways it can be factored:
1494 = 2(747)(1),
1494 = 2(249)(3), and
1494 = 2(83)(9).
And because for whole numbers where a > b, 2(a)(b), a² – b², a² + b² will be a Pythagorean triple.

1483 and Level 5

Today’s Puzzle:

Some of the clues in the same row or column in this puzzle have more than one common factor. In each case, will you make the logical choice to find the puzzle’s unique solution?

Factors of 1483:

  • 1483 is a prime number.
  • Prime factorization: 1483 is prime.
  • 1483 has no exponents greater than 1 in its prime factorization, so √1483 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1483 has exactly 2 factors.
  • The factors of 1483 are outlined with their factor pair partners in the graphic below.

How do we know that 1483 is a prime number? If 1483 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1483. Since 1483 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1483 is a prime number.

More about the Number 1483:

1483 is the difference of two squares:
742² – 741² = 1483

The first five prime decades are listed below. 1483 is the second prime number in the fifth prime decade: