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

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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!

1450 A Pair of Factor Trees

On today’s puzzle, there are two small Christmas trees. Will two smaller trees on the puzzle be easier to solve than one big one? You’ll have to try it to know!

Every puzzle has a puzzle number to distinguish it from the others. Here are some facts about this puzzle number, 1450:

  • 1450 is a composite number.
  • Prime factorization: 1450 = 2 × 5 × 5 × 29, which can be written 1450 = 2 × 5² × 29
  • 1450 has at least one exponent greater than 1 in its prime factorization so √1450 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1450 = (√25)(√58) = 5√58
  • 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 1450 has exactly 12 factors.
  • The factors of 1450 are outlined with their factor pair partners in the graphic below.

In case you are looking for factor trees for 1450, here are two different ones:

1450 is the hypotenuse of SEVEN Pythagorean triples:
170-1440-1450 which is 10 times (17-144-145)
240-1430-1450 which is 10 times (24-143-145)
406-1392-1450 which is (7-24-25) times 58
666-1288-1450 which is 2 times (333-644-725)
728-1254-1450 which is 2 times (364-627-725)
870-1160-1450 which is (3-4-5) times 290
1000-1050-1450 which is (20-21-29) times 50

1437 Belt Buckle Mystery

I have childhood memories of preparing for Thanksgiving at school by making pilgrim hats, pilgrim shirts, or pilgrim shoes out of construction paper. Each of those clothing items had a distinguished buckle. Why? That buckle’s popularity was a mystery to me until I read an article by Ken Jennings titled The Debunker: What Did Pilgrim Hats Really Look Like?

This mystery level puzzle has a buckle, too. Perhaps it can help debunk the idea that math is no fun while it helps you learn about factors or reinforces your memory of the multiplication table.

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

The puzzle number is 1437. Here are a few facts about that number:

  • 1437 is a composite number.
  • Prime factorization: 1437 = 3 × 479
  • 1437 has no exponents greater than 1 in its prime factorization, so √1437 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 1437 has exactly 4 factors.
  • The factors of 1437 are outlined with their factor pair partners in the graphic below.

1437 is the difference of two squares in two different ways:
719² – 718² = 1437
241² – 238² = 1437

1436 This Turkey Is Ready

If this year you find yourself needing a bigger turkey, then this one fits that description. This turkey uses math facts from a 14 × 14 multiplication table.

Adding prime number 13 doesn’t cause any problems. You hopefully will recognize that 39 and 119 are both multiples of 13.

However, adding 14 to the puzzle might make you want to call “fowl” because clues 28, 56, 70, and 84 all have two sets of factor pairs where both factors are 14 or less. Can you figure out which clues use 7 and which use 14?

Like always, there is only one solution.

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

I enjoyed this tweet from Robin Schwartz. Spoiler alert: If you click on the link, you will see the solution.

That was puzzle number 1436. Here are a few facts about that number:

  • 1436 is a composite number.
  • Prime factorization: 1436 = 2 × 2 × 359, which can be written 1436 = 2² × 359
  • 1436 has at least one exponent greater than 1 in its prime factorization so √1436 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1436 = (√4)(√359) = 2√359
  • 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 1436 has exactly 6 factors.
  • The factors of 1436 are outlined with their factor pair partners in the graphic below.

1436 is the difference of two squares:
360² – 358² = 1436

 

1427 Mysterious Cat

This mysterious Halloween cat shares twelve clues that can help you solve its puzzle. Each clue is the products of two factors from 1 to 12 multiplied together. Will you be able to solve its mystery?

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you a little bit about the puzzle number, 1427:

  • 1427 is a prime number.
  • Prime factorization: 1427 is prime.
  • 1427 has no exponents greater than 1 in its prime factorization, so √1427 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1427 has exactly 2 factors.
  • The factors of 1427 are outlined with their factor pair partners in the graphic below.

How do we know that 1427 is a prime number? If 1427 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1427. Since 1427 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1427 is a prime number.

1427 is 272 in BASE 25 because 2(25²) + 7(25) + 2(1) = 1427