1545 The Complete Message Made with Puzzle Letters

The Puzzles:

The last six Find the Factors Puzzles spelled out this message: I voted. I like the quote from Larry J. Sabato, “Every election is determined by the people who show up.” My vote will matter, and so will yours.

You can also print these puzzles from this excel file: Taxman & 1537-1544

American politics has become rather ugly, but voting is still important.

I am thrilled that I was able to vote for one of these candidates for governor of Utah, and I am very happy that one of them will become my governor even if it isn’t the same person I voted for.

Factors of 1545:

  • 1545 is a composite number.
  • Prime factorization: 1545 = 3 × 5 × 309.
  • 1545 has no exponents greater than 1 in its prime factorization, so √1545 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 1545 has exactly 8 factors.
  • The factors of 1545 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1545:

1545 is the hypotenuse of a Pythagorean triple:
927-1236-1545, which is (3-4-5) times 309.

1544 Final Letter of the Message

Today’s Puzzle:

This is the sixth and final letter of the message that I made for you. It’s a mystery puzzle, but that doesn’t necessarily mean that’s it’s a difficult puzzle. Give it a try and think about why I sent the message.

Factors of 1544:

  • 1544 is a composite number.
  • Prime factorization: 1544 = 2 × 2 × 2 × 193, which can be written 1544 = 2³ × 193
  • 1544 has at least one exponent greater than 1 in its prime factorization so √1544 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1544 = (√4)(√386) = 2√386
  • 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 1544 has exactly 8 factors.
  • The factors of 1544 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1544:

1544 is the sum of two squares:
38² + 10² = 1544.

1544 is the hypotenuse of a Pythagorean triple:
760-1344-1544, calculated from 2(38)(10), 38² – 10², 38² + 10².
It is also 8 times (95-168-193).

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.)

1542 Fourth Letter of the Message

Today’s Puzzle:

This level one puzzle is the easiest of all the puzzles in my message to you. It might also be the letter that helps you know exactly what the message says.

Factors of 1542:

  • 1542 is a composite number.
  • Prime factorization: 1542 = 2 × 3 × 257.
  • 1542 has no exponents greater than 1 in its prime factorization, so √1542 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 1542 has exactly 8 factors.
  • The factors of 1542 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1542

1542 is the hypotenuse of a Pythagorean triple:
192-1530-1542, which is 6 times (32-255-257).

 

1541 is the 23rd Octagonal Number

Today’s Puzzle:

Into which geometric shape can you arrange 1541 tiny squares?

The answer is an octagon as illustrated below.

Can you divide that octagon into 6 triangles, 5 of them representing the 22nd triangular number and 1 of them representing the 23rd?

It is possible because 5(22·23)/2 + (23·24)/2 = 1541.

We can simplify the left side of the equal sign:
5(22·23)/2 + (23·24)/2 =
5(11·23) + (23·12) =
23(5·11) + 23(12) =
23(55 + 12) =
23(67) = 1541

Factors of 1541:

  • 1541 is a composite number.
  • Prime factorization: 1541 = 23 × 67.
  • 1541 has no exponents greater than 1 in its prime factorization, so √1541 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 1541 has exactly 4 factors.
  • The factors of 1541 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1541:

1541 is the difference of two squares in two different ways:
771² – 770² = 1541 (Not coincidently, 770 + 771 = 1541), and
45² – 22² = 1541.

1539 Mystery Puzzle for a Mystery Message

Today’s Puzzle:

This is the third letter in my message to you. Have you figured out the complete message yet?

The puzzle itself is a mystery level puzzle. Will it be easy or difficult to do? You’ll have to try it to know the answer to that one!

Factors of 1539:

Adding up the non-nine digits we get 1 + 5 + 3 = 9, so 1539 is divisible by 9. In fact, it is divisible by 9 twice!

  • 1539 is a composite number.
  • Prime factorization: 1539 = 3 × 3 × 3 × 3 × 19, which can be written 1539 = 3⁴ × 19
  • 1539 has at least one exponent greater than 1 in its prime factorization so √1539 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1539 = (√81)(√19) = 9√19
  • 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 1539 has exactly 10 factors.
  • The factors of 1539 are outlined with their factor pair partners in the graphic below.

More about the Number 1539:

1539 is the difference of two squares in five different ways:
770² – 769² = 1539,
258² – 255² = 1539,
90² – 81² = 1539,
50² – 31² = 1539, and
42² – 15² = 1539.

1538 Second Letter of My Message

Today’s Puzzle:

This puzzle is the second letter in a message I want to give you. I will give more letters in a few days. How many letters will you need to figure out the message?

I gave you the clues of this level 3 multiplication table puzzle in an order that makes finding the solution easier. It can be solved by finding the common factor of the two clues in the first row of the puzzle and then working your way down the puzzle row by row until you have found all the factors from 1 to 12 for both the first column and the top row.

Factors of 1538:

  • 1538 is a composite number.
  • Prime factorization: 1538 = 2 × 769.
  • 1538 has no exponents greater than 1 in its prime factorization, so √1538 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 1538 has exactly 4 factors.
  • The factors of 1538 are outlined with their factor pair partners in the graphic below.

More about the Number 1538:

1538 is the sum of two squares:
37² + 13² = 1538.

1538 is the hypotenuse of a Pythagorean triple:
962-1200-1538 calculated from 2(37)(13), 37² – 13², 37² + 13².

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.

 

 

Factors of 1536 Make Sum-Difference!

Today’s Puzzles:

1536 has 10 different factor pairs. One of those pairs adds up to 80 and a different one subtracts to give 80. Can you find the factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1536 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.

That first puzzle is the first possible Sum-Difference puzzle. The second puzzle is only the 49th possible puzzle.

Factors of 1536:

  • 1536 is a composite number.
  • Prime factorization: 1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3, which can be written 1536 = 2⁹ × 3.
  • 1536 has at least one exponent greater than 1 in its prime factorization so √1536 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1536 = (√256)(√6) = 16√6
  • The exponents in the prime factorization are 9 and 1. Adding one to each exponent and multiplying we get (9 + 1)(1 + 1) = 10 × 2 = 20. Therefore 1536 has exactly 20 factors.
  • The factors of 1536 are outlined with their factor pair partners in the graphic below.

More About the Number 1536:

1536 is the difference of two squares in EIGHT different ways!
385² – 383² = 1536,
194² – 190² = 1536,
131² – 125² = 1536,
100² – 92² = 1536,
70² – 58² = 1536,
56² – 40² = 1536,
44² – 20² = 1536, and
40² – 8² = 1536.

1536 = 6 × 256.
256 is the 100th number whose square root can be simplified.
1536 is the 600th number whose square root can be simplified.

Here are the 501st to the 600th simplifiable square roots. If at least three square roots are consecutive, they are highlighted.

 

1535 Your Favorite Candy Bar

Today’s Puzzle:

If you could have your favorite candy bar in the largest size possible, would it actually turn out to be more than you can chew? Of course not. Even if it took a long time for you to finish that candy bar, it would not be an impossible challenge.

Think of this Challenge puzzle the same way. I think this one uses some delightful logic. Find that logic, take your time, and enjoy the process!

Print the puzzles or type the solution in this excel file: 10 Factors 1526-1535

Factors of 1535:

  • 1535 is a composite number.
  • Prime factorization: 1535 = 5 × 307.
  • 1535 has no exponents greater than 1 in its prime factorization, so √1535 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 1535 has exactly 4 factors.
  • The factors of 1535 are outlined with their factor pair partners in the graphic below.

Another Fact about the number 1535:

1535 is the hypotenuse of a Pythagorean triple:
921-1228-1535 which is (3-4-5) times 307.