Puzzles

1511 and Level 1

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Today’s Puzzle:

Do you know what number is the common factor of all the clues in this puzzle? If you do, then you are well on your way to solving this level one puzzle!

Factors of 1511:

  • 1511 is a prime number.
  • Prime factorization: 1511 is prime.
  • 1511 has no exponents greater than 1 in its prime factorization, so √1511 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1511 has exactly 2 factors.
  • The factors of 1511 are outlined with their factor pair partners in the graphic below.

How do we know that 1511 is a prime number? If 1511 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1511. Since 1511 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1511 is a prime number.

One More Fact about the Number 1511:

1511 is equidistant from the two closest primes:
The previous prime was 1511 – 12 = 1499.
The next prime will be 1511 + 12 = 1523.

1510 Challenge Puzzle

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Today’s Puzzle:

Challenge puzzles are like four multiplication tables connected to each other. Use logic to place the factors 1 to 10 in each boldly outlined column or row so that the given clues are the products of the factors you write. I hope you enjoy solving this puzzle as much as I enjoyed making it for you!

Here’s an excel file with this week’s puzzles: 10 Factors 1502-1510

Factors of 1510:

  • 1510 is a composite number.
  • Prime factorization: 1510 = 2 × 5 × 151.
  • 1510 has no exponents greater than 1 in its prime factorization, so √1510 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 1510 has exactly 8 factors.
  • The factors of 1510 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1510:

1510 is the hypotenuse of a Pythagorean triple:
906-1208-1510, which is (3-4-5) times 302.

 

1509 A Mystery Level Puzzle on Training Wheels

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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

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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.

 

1507 and Level 6

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Today’s Puzzle:

Level six puzzles are designed to be tricky, but if you examine the clues, there is a logical place to begin, and logic can help you complete the entire puzzle.

Factors of 1507:

  • 1507 is a composite number.
  • Prime factorization: 1507 = 11 × 137.
  • 1507 has no exponents greater than 1 in its prime factorization, so √1507 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 1507 has exactly 4 factors.
  • The factors of 1507 are outlined with their factor pair partners in the graphic below.

More about the Number 1507:

1507 is the hypotenuse of a Pythagorean triple.
968-1155-1507

A divisibility trick tells us that all of the numbers in that triple are divisible by 11:
9 – 6 + 8 = 11,
1 – 1 + 5 – 5 = 0,
1 – 5 + 0 – 7 = -11.

Yes, you would need to understand negative numbers for that last one, but 11, 0, and -11 can all be evenly divided by 11 so the corresponding numbers are also divisible by 11.

In fact, 968-1155-1507 is just 11 times (88-105-137).

1506 and Level 5

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Today’s Puzzle:

Can you find the factors 1 to 10 in a logical order so that the given clues are the products of those factors? Don’t let any of the clues trick you!

Factors of 1506:

  • 1506 is a composite number.
  • Prime factorization: 1506 = 2 × 3 × 251.
  • 1506 has no exponents greater than 1 in its prime factorization, so √1506 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 1506 has exactly 8 factors.
  • The factors of 1506 are outlined with their factor pair partners in the graphic below.

Pythagorean triples with 1506:

1506 is not the sum or the difference of two squares but it is still part of two Pythagorean triples:
1506-567008-567010, calculated from 2(753)(1), 753² – 1², 753² + 1², and
1506-62992-63010, calculated from 2(251)(3), 251² – 1², 251² + 1².

1504 and Level 3

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Today’s Puzzle:

Since this is a level 3 puzzle the clues are given in a logical order from top to bottom. Write the factors 1 to 10 in the first column and again in the top row.

Usually, you only have to consider the previous clues when finding the factors in a level 3 puzzle, but when you consider if 4 = 2 × 2 or 1 × 4, you will also have to look at a clue below it. You can do this!

Factors of 1504:

  • 1504 is a composite number.
  • Prime factorization: 1504 = 2 × 2 × 2 × 2 × 2 × 47, which can be written 1504 = 2⁵ × 47.
  • 1504 has at least one exponent greater than 1 in its prime factorization so √1504 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1504 = (√16)(√94) = 4√94.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1504 has exactly 12 factors.
  • The factors of 1504 are outlined with their factor pair partners in the graphic below.

More about the Number 1504:

1504 is the difference of two squares in four different ways:
377² – 375² = 1504
190² – 186² = 1504
98² – 90² = 1504
55² – 39² = 1504

Why is 1503 a Friedman Number?

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Friedman Puzzle:

Can you find an expression equaling 1503 that uses 1, 5, 0, and 3 each exactly once, but in any order, and some combination of  +, -, ×, or ÷? For this particular Friedman puzzle, none of those digits are exponents. If you can solve this Friedman puzzle, you will know why 1503 is the 24th Friedman number. You can find the solution hidden someplace in this post. (By the way, another permutation of those digits, 1530, will be the 25th Friedman number!)

Find the Factors Puzzle:

There are 14 clues in this level 2 puzzle. Use those clues and logic to place the factors 1 to 10 in both the first column and the top row. That’s how you start to turn this puzzle into a multiplication table!

Factors of 1503:

  • 1503 is a composite number.
  • Prime factorization: 1503 = 3 × 3 × 167, which can be written 1503 = 3² × 167.
  • 1503 has at least one exponent greater than 1 in its prime factorization so √1503 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1503 = (√9)(√167) = 3√167.
  • 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 1503 has exactly 6 factors.
  • The factors of 1503 are outlined with their factor pair partners in the graphic below.

Did you see the solution to the Friedman puzzle in that factor pair chart?

1502 and Level 1

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Today’s Puzzle:

This level 1 puzzle has products in one of the rows and in one of the columns. Can you use those products to figure out where the factors 1 to 10 belong in this multiplication table puzzle?

Factors of 1502:

  • 1502 is a composite number.
  • Prime factorization: 1502 = 2 × 751.
  • 1502 has no exponents greater than 1 in its prime factorization, so √1502 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 1502 has exactly 4 factors.
  • The factors of 1502 are outlined with their factor pair partners in the graphic below.

Pythagorean Triple with 1502:

1502 is not the sum or the difference of two squares, but 1502 = 2(751)(1), so it is part of a Pythagorean triple:
1502-564000-564002, calculated from 2(751)(1), 751² – 1², 751² – 1².

1499 Challenge Puzzle

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Today’s Puzzle:

Use the 19 clues, logic, and the multiplication facts from a 10 × 10 multiplication table to find the unique solution of this Find the Factors Challenge puzzle. Good luck!

Factors of 1499:

  • 1499 is a prime number.
  • Prime factorization: 1499 is prime.
  • 1499 has no exponents greater than 1 in its prime factorization, so √1499 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1499 has exactly 2 factors.
  • The factors of 1499 are outlined with their factor pair partners in the graphic below.

How do we know that 1499 is a prime number? If 1499 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1499. Since 1499 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1499 is a prime number.

More Facts about the Number 1499:

1499 is the difference of two consecutive squares:
750² – 749² = 1499

oeis.org reminds us that 149, 199, and 499 are also prime numbers, so taking away one digit from 1499 always leaves a prime number.