# 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).

# 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.

# 1534 Dem Bones

### Today’s Puzzle:

You’ve probably heard someone sing about Dem Bones before. It is a song that describes how bones in the body are connected to each other. Ezekiel 37:1-14 was the inspiration for the lyrics, and the song is the inspiration for today’s mystery level puzzle: ### Factors of 1534:

• 1534 is a composite number.
• Prime factorization: 1534 = 2 × 13 × 59.
• 1534 has no exponents greater than 1 in its prime factorization, so √1534 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 1534 has exactly 8 factors.
• The factors of 1534 are outlined with their factor pair partners in the graphic below. ### More Facts about the Number 1534:

1534 is the hypotenuse of a Pythagorean triple:
590-1416-1534 which is (5-12-13) times 118.

1534 is 4(7³) + 3(7²) + 2(7¹) + 1(7º). That means it is 4321 in base 7.

# 1533 Double Double Toil and Trouble

### Today’s Puzzle:

Solving this cauldron-looking mystery level puzzle might be double-double toil and trouble.  Why? There are four sets of clues that have more than one possible common factor. Nevertheless, the puzzle has only one solution, and you can find it if you let logic and a basic multiplication table be your guide. ### Factors of 1533:

Double Double Toil and Trouble… You can use two divisibility tricks to find the factors of 1533, and the second trick uses doubling.

Is 1533 divisible by 3? The trick for finding numbers divisible by 3 is to add up their digits. If the sum is divisible by 3, so is the original number:
1 + 5 + 3 + 3 = 12, so 1533 is divisible by 3.
It really is! See what I mean: 1533 ÷ 3 = 511.

Is it divisible by 7? Look at the previous quotient, 511. Separate the last digit from the rest and subtract its double from the first part: 51 – 2 = 49, so 511 is divisible by 7, and its multiple, 1533, is also divisible by 7.

(We could have used the same doubling trick TWICE on 1533, but that definitely would have been double-double, toil, and trouble. Just dividing 1533 by 7 to get 219 would be easier!)

• 1533 is a composite number.
• Prime factorization: 1533 = 3 × 7 × 73.
• 1533 has no exponents greater than 1 in its prime factorization, so √1533 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 1533 has exactly 8 factors.
• The factors of 1533 are outlined with their factor pair partners in the graphic below. ### More Facts About the Number 1533:

1533 is the hypotenuse of a Pythagorean triple, and you can apply those same two divisibility tricks to all three numbers in that triple. (Note: 3 × 7 = 21)
1008-1155-1533 which is 21 times (48-55-73)

1533 is the difference of two squares in FOUR different ways:
767² – 766² = 1533,
257² – 254² = 1533,
113² – 106² = 1533, and
47² – 26² = 1533.

# 1532 Don’t Let This Puzzle Spook You!

### Today’s Puzzle:

If this little ghost prowls your neighborhood this Halloween, don’t let it spook you. Sure, it is a level 6 puzzle, but if you stick to using logic from start to finish, you’ll know the most about this ghost! Here’s the same puzzle minus the embellishments: ### Factors of 1532:

• 1532 is a composite number.
• Prime factorization: 1532 = 2 × 2 × 383, which can be written 1532 = 2² × 383
• 1532 has at least one exponent greater than 1 in its prime factorization so √1532 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1532 = (√4)(√383) = 2√383
• 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 1532 has exactly 6 factors.
• The factors of 1532 are outlined with their factor pair partners in the graphic below. ### More about the Number 1532:

1532 = 2 × 383 × 2. That factorization looks the same frontwards or backward.

1532 can be written as the difference of two squares:
384² – 382² = 1532.