Puzzles

1542 Fourth Letter of the Message

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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 = 1¹ + 5¹ + 4⁵ + 2⁹ so it is a digitally powerful number.

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

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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

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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

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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.

 

 

1535 Your Favorite Candy Bar

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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.

1533 Double Double Toil and Trouble

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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!

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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:

If you know what a normal distribution is, then you should enjoy this statistics joke:

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.

 

 

1530 Jack-o’-lantern

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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.

 

 

1529 Pointy Hat

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Today’s Puzzle:

A pointy hat is part of many different Halloween costumes: Wizards, Witches, Medieval Princesses, and Clowns come to my mind. Today’s puzzle looks like a pointy hat. Use logic to work your magic in solving it! As always, there is only one solution.

Factors of 1529:

Look at this math fact using the digits of 1529:
1 – 5 + 2 – 9 = -11, a number divisible by 11, so 1529 is divisible by 11.

  • 1529 is a composite number.
  • Prime factorization: 1529 = 11 × 139.
  • 1529 has no exponents greater than 1 in its prime factorization, so √1529 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 1529 has exactly 4 factors.
  • The factors of 1529 are outlined with their factor pair partners in the graphic below.

More about the Number 1529:

1529 is the difference of two squares in two different ways:
765² – 764²  = 1529, and
75² – 64²  = 1529.

 

1528 Candy Corn

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Today’s Puzzle:

To solve this Level 3 Candy Corn Halloween puzzle, first, find the factors that will work with the clues in the top and bottom rows. Then work you way down row by row filling in factors as you go.

When you get to the 8 in this puzzle, will the factors be 8 × 1 or 4 × 2? Two of those factors will be eliminated because they already appear in the first column. The other two remain possibilities, but one of those factors cannot appear in any other place in that first column, so that is the one you will want to choose. Have a sweet time solving this puzzle!

Here is a plain version of the same puzzle:

Factors of 1528:

1528 is divisible by two because it is even.

1528 is divisible by four because its last two digits (in the same order) make a number, 28, which is divisible by 4.

Can 1528 be evenly divided by 8? Yes. Here’s a quick way to know: 28 is divisible by 4, but not by 8, AND 5 is odd, so 1528 is divisible by 8, as is every other number ending in 528.

  • 1528 is a composite number.
  • Prime factorization: 1528 = 2 × 2 × 2 × 191, which can be written 1528 = 2³ × 191.
  • 1528 has at least one exponent greater than 1 in its prime factorization so √1528 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1528 = (√4)(√382) = 2√382.
  • 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 1528 has exactly 8 factors.
  • The factors of 1528 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1528:

Since 1528 is divisible by 8 but not by 16, it can be written as the sum of 16 consecutive numbers:
88+89+90+91+92+93+94+95+96+97+98+99+100+101+102+103=1528.

Note that 95 + 96 = 191, and 8 × 191 = 1528.
Likewise, 94 + 97 = 1528,
93 + 98 = 1528, and so forth until we get to…
88 + 103 = 1528.