1509 A Mystery Level Puzzle on Training Wheels

Today’s Puzzle:

I have described level 3 puzzles as level 4 puzzles on training wheels. Today’s puzzle is definitely not a level 3 puzzle, but it is on training wheels. A logical way to solve this puzzle is to start with the clue in the top row and work your way down the puzzle row by row writing the factors as you go. Giving you the logical order to use the clues should help some, but the logic needed to find the factors will still be a mystery. Don’t guess and check. Please, use logic! Think about how each clue relates to the other clues in the puzzle.

Factors of 1509:

  • 1509 is a composite number.
  • Prime factorization: 1509 = 3 × 503
  • 1509 has no exponents greater than 1 in its prime factorization, so √1509 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 1509 has exactly 4 factors.
  • The factors of 1509 are outlined with their factor pair partners in the graphic below.

More about the Number 1509:

1509 is the difference of two squares in two different ways:
755² – 754² = 1509
253² – 250² = 1509

1508 Hosting a Playful Math Carnival and Flying by the Seat of My Pants

Blog Submission Appeal:

Please, tell me how you’ve made K-12 math education more fun. You see, later this month I’m hosting the Playful Math Education Blog Carnival. I have found several great blog posts to share, but maybe I haven’t seen yours. You can share your blog post with me by submitting this official form, leaving a comment below, or messaging me on twitter, Iva Sallay@findthefactors.com. I look forward to reading your post! Please share it with me by Saturday, August 22 so it can be included in this month’s carnival.

Today’s Puzzle:

This mystery-level puzzle was modeled after a carnival ride, the swing carousel, a ride that tilts slightly as it goes around, and lets you ride by the seat of your pants. My puzzle might not be the best representation of that ride, but it hopefully got your attention. 

Embellishing the puzzle might make it more eye-catching, but it is probably easier to solve the puzzle without distracting color and lines. (It’s a mystery-level puzzle, so I’m keeping how easy or difficult it is a secret.) Here is a plain version of the same puzzle:

Factors of 1508:

  • 1508 is a composite number.
  • Prime factorization: 1508 = 2 × 2 × 13 × 29, which can be written 1508 = 2² × 13 × 29
  • 1508 has at least one exponent greater than 1 in its prime factorization so √1508 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1508 = (√4)(√377) = 2√377
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1508 has exactly 12 factors.
  • The factors of 1508 are outlined with their factor pair partners in the graphic below.

More about the Number 1508:

1508 is the sum of two squares in two different ways:
32² + 22² = 1508
38² + 8² = 1508

1508 is the hypotenuse of FOUR Pythagorean triples:
540-1408-1508, calculated from 32² – 22², 2(32)(22), 32² + 22²,
580-1392-1508, which is (5-12-13) times 116,
608-1380-1508, calculated from 2(38)(8), 38² – 8², 38² + 8²,
1040-1092-1508 which is (20-21-29) times 52.

 

1498 Another Mystery

Today’s Puzzle:

The number 48 appears four times in a 12 × 12 multiplication table, and all four 48’s appear in this puzzle! Where will you put its factors, 6, 8, 4, 12? Use logic to figure out where all the numbers from 1 to 12 need to go to make this puzzle turn into a multiplication table.

Factors of 1498:

  • 1498 is a composite number.
  • Prime factorization: 1498 = 2 × 7 × 107.
  • 1498 has no exponents greater than 1 in its prime factorization, so √1498 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 1498 has exactly 8 factors.
  • The factors of 1498 are outlined with their factor pair partners in the graphic below.

Pythagorean Triples with 1498:

1498 is 2(107)(7) so we can calculate a cool Pythagorean triple from
2(107)(7), 107² – 7², 107² + 7² to get 1498-11400-11498.

Also from 2(749)(1), 749² – 1², 749² + 1², we get 1498-561000-561002.

 

1497 Mystery

Today’s Puzzle:

Knowing where to place some of the numbers from 1 to 12 in this puzzle shouldn’t be too difficult, but placing ALL of the numbers will be like solving a mystery.

Factors of 1497:

  • 1497 is a composite number.
  • Prime factorization: 1497 = 3 × 499
  • 1497 has no exponents greater than 1 in its prime factorization, so √1497 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 1497 has exactly 4 factors.
  • The factors of 1497 are outlined with their factor pair partners in the graphic below.

More about the Number 1497:

1497 is the difference of two squares in two different ways:
749² – 748² = 1497
251² – 248² = 1497

1486 Mysterious Cat

Today’s Puzzle:

If a cat has nine lives, how many lives do seven cats have? Where do the numbers 9 and 7 belong in this puzzle? Where do all the other numbers from 1 to 10 belong?

Factors of 1486:

  • 1486 is a composite number.
  • Prime factorization: 1486 = 2 × 743
  • 1486 has no exponents greater than 1 in its prime factorization, so √1486 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 1486 has exactly 4 factors.
  • The factors of 1486 are outlined with their factor pair partners in the graphic below.

More about the Number 1486:

1486 is not the sum of two squares or the difference of two squares, but it is in a Pythagorean triple:
1486-552048-552050, calculated from 2(743)(1), 743² – 1², 743² + 1²

1485 Sticky Lollipop Mystery

Today’s Puzzle:

Lollipops can be very sticky. Will the logic needed for this puzzle be a sticky mess, or will you be able to lick it? That’s the mystery. Good luck!

Factors of 1485:

  • 1485 is a composite number.
  • Prime factorization: 1485 = 3 × 3 × 3 × 5 × 11, which can be written 1485 = 3³ × 5 × 11
  • 1485 has at least one exponent greater than 1 in its prime factorization so √1485 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1485 = (√9)(√165) = 3√165
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1485 has exactly 16 factors.
  • The factors of 1485 are outlined with their factor pair partners in the graphic below.

More about the Number 1485:

Did you notice: (27)(55) = 1485? That means that
(54)(55)/2 = 1485 so 1485 is the 54th triangular number.

1485 is the hypotenuse of a Pythagorean triple:
891-1188-1485 which is (3-4-5) times 297.

Of the ten numbers from 1480 to 1489, four are prime numbers and have exactly 2 factors. Three of the ten numbers have exactly 16 factors, namely 1480, 1482, and 1485. No smaller set of three numbers with sixteen factors are as close together as these three are!

1476 Mystery

Today’s Puzzle:

To solve this mystery puzzle, first, gather some facts. I mean, common factors:
Common Factors of 6, 3, 12, and 9 are 1 and 3.
For 20 and 10 we have 2, 5, and 10.
And for 20 and 60, we can only use 5 or 10 because common factor, 20, is too big for a 1 to 12 puzzle.

Which common factors “Done it”? Don’t jump to conclusions. Remember, a good detective will use the facts and logic to figure out the mystery. Good luck!

Factors of 1476:

  • 1476 is a composite number.
  • Prime factorization: 1476 = 2 × 2 × 3 × 3 × 41, which can be written 1476 = 2² × 3² × 41
  • 1476 has at least one exponent greater than 1 in its prime factorization so √1476 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1476 = (√36)(√41) = 6√41
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1476 has exactly 18 factors.
  • The factors of 1476 are outlined with their factor pair partners in the graphic below.

Simplifying √1476 doesn’t have to be a mystery! Here’s one strategy to do it:

More Facts about the number 1476:

1476 is the sum of two squares:
30² + 24² = 1476

1476 is the hypotenuse of a Pythagorean triple:
324-1440-1476 which is 36 times (9-40-41),
and can also be calculated from 30² – 24², 2(30)(24), 30² + 24².

 

1475 The Logic Needed to Solve This Mystery

Today’s Puzzle:

Solving this puzzle will be as easy as 1-2-3 if you begin by asking yourself which clues must use the 1’s, the 2’s, and the 3’s, but not necessarily in that order. At the same time, you can eliminate two common factors of 48 and 72, and finishing the puzzle should then become fairly routine.

Factors of 1475:

  • 1475 is a composite number.
  • Prime factorization: 1475 = 5 × 5 × 59, which can be written 1475 = 5² × 59
  • 1475 has at least one exponent greater than 1 in its prime factorization so √1475 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1475 = (√25)(√59) = 5√59
  • 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 1475 has exactly 6 factors.
  • The factors of 1475 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1475:

1475 is the hypotenuse of two Pythagorean triples:
413-1416-1475 which is (7-24-25) times 59.
885-1180-1475 which is (3-4-5) times 295.

 

1464 Alibis and a Mystery

Today’s Puzzle:

You find clues 30, 30, 30, 54. Two of those 30’s claim to be 3 × 10.
When you find clues 40, 40, 56, you realize that at least one of the 40’s must be 4 x 10.

Can you believe the alibis the 30’s have just given you?
Can you put your detective skills together to figure out this mystery?

Factors of 1464:

  • 1464 is a composite number.
  • Prime factorization: 1464 = 2 × 2 × 2 × 3 × 61, which can be written 1464 = 2³ × 3 × 61
  • 1464 has at least one exponent greater than 1 in its prime factorization so √1464 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1464 = (√4)(√366) = 2√366
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1464 has exactly 16 factors.
  • The factors of 1464 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1464:

1464 is the hypotenuse of a Pythagorean triple:
264-1440-1464 which is 24 times (11-60-61).

Stetson.edu alerts us that 1464 is a repdigit in two different bases:
It’s 1111 in BASE 11 because 11³ + 11² + 11¹ + 11º = 1464, and
it’s 888 in BASE 13 because 8(13² + 13¹ + 13º) = 8(183) =1464.

1452 Poinsettia Plant Mystery

Merry Christmas, Everybody!

The poinsettia plant has a reputation for being poisonous, but it has never been a part of a whodunnit, and it never will. Poinsettias actually aren’t poisonous.

Multiplication tables might also have a reputation for being deadly, but they aren’t either, except maybe this one. Can you use logic to solve this puzzle without it killing you?

To solve the puzzle, you will need some multiplication facts that you probably DON’T have memorized. They can be found in the table below. Be careful! The more often a clue appears, the more trouble it can be:

Notice that the number 60 appears EIGHT times in that table. Lucky for you, it doesn’t appear even once in today’s puzzle!

Now I’d like to factor the puzzle number, 1452. Here are a few facts about that number:

1 + 4 + 5 + 2 = 12, which is divisible by 3, so 1452 is divisible by 3.
1 – 4 + 5 – 2 = 0, which is divisible by 11, so 1452 is divisible by 11.

  • 1452 is a composite number.
  • Prime factorization: 1452 = 2 × 2 × 3 × 11 × 11, which can be written 1452 = 2² × 3 × 11²
  • 1452 has at least one exponent greater than 1 in its prime factorization so √1452 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1452 = (√484)(√3) = 22√3
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 1452 has exactly 18 factors.
  • The factors of 1452 are outlined with their factor pair partners in the graphic below.

To commemorate the season, here’s a factor tree for 1452:

Have a very happy holiday!