Prime factorization

1518 and Level 6

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Today’s Puzzle:

Level 6 puzzles are designed to be a little tricky. Just make sure you use logic to figure out the factors every time, and you will get it done!

Factors of 1518:

  • 1518 is a composite number.
  • Prime factorization: 1518 = 2 × 3 × 11 × 23.
  • 1518 has no exponents greater than 1 in its prime factorization, so √1518 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1518 has exactly 16 factors.
  • The factors of 1518 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1518:

1518 has a palindromic prime factorization. (The digits are the same frontward or backward.)
1518 = 2 · 3 · 11 · 23

1517 and Level 5

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Today’s Puzzle:

Which common factor of 72 and 36 is needed to solve this puzzle? Is it 6, 9, or 12? There is an easier place to begin this level 5 puzzle. Don’t guess and check. Use logic to know which factors you should use.  You can figure it out!

Factors of 1517:

  • 1517 is a composite number.
  • Prime factorization: 1517 = 37 × 41.
  • 1517 has no exponents greater than 1 in its prime factorization, so √1517 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 1517 has exactly 4 factors.
  • The factors of 1517 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1517:

1517 is the difference of two squares in two different ways:
759² – 758² = 1517,
39² – 2² = 1517.

1517 is also the sum of two squares in two different ways:
34² + 19² = 1517,
29² + 26² = 1517.

1517 is the hypotenuse of FOUR Pythagorean triples:
165-1508-1517, calculated from 29² – 26², 2(29)(26), 29² + 26²,
333-1480-1517, which is 37 times (9-40-41),
492-1435-1517, which is (12-35-37) times 41,
795-1292-1517, calculated from 34² – 19², 2(34)(19), 34² + 19².

1516 and Level 4

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Today’s Puzzle:

Using logic, write each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.

Factors of 1516:

  • 1516 is a composite number.
  • Prime factorization: 1516 = 2 × 2 × 379, which can be written 1516 = 2² × 379.
  • 1516 has at least one exponent greater than 1 in its prime factorization so √1516 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1516 = (√4)(√379) = 2√379.
  • 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 1516 has exactly 6 factors.
  • The factors of 1516 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1516:

Only one of its factor pairs add up to an even number, so 1516 is the difference of two squares in only one way:
380² – 378² = 1516.

1515 and Level 3

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Today’s Puzzle:

Start with the greatest common factor of 30 and 48, write the factors in the appropriate boxes, then work your way down this level 3 puzzle row by row using logic until you have found all the factors. You can do this!

Factors of 1515:

  • 1515 is a composite number.
  • Prime factorization: 1515 = 3 × 5 × 101.
  • 1515 has no exponents greater than 1 in its prime factorization, so √1515 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 1515 has exactly 8 factors.
  • The factors of 1515 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1515:

1515 is the hypotenuse of FOUR Pythagorean triples:
300-1485-1515, which is 15 times (20-99-101),
651-1368-1515, which is 3 times (217-456-505),
909-1212-1515, which is (3-4-5) times 303,
1008-1131-1515, which is 3 times (336-377-505).

1514 and Level 2

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Today’s Puzzle:

There is only one way to write the factors from 1 to 12 in both the first column and the top row so that this puzzle will behave like a multiplication table. The given clues will be the products of the factors you write. Can you find the way?

Factors of 1514:

  • 1514 is a composite number.
  • Prime factorization: 1514 = 2 × 757.
  • 1514 has no exponents greater than 1 in its prime factorization, so √1514 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 1514 has exactly 4 factors.
  • The factors of 1514 are outlined with their factor pair partners in the graphic below.

More Facts about the number 1514:

1514 is the sum of two squares:
35² + 17² = 1514

1514 is the hypotenuse of a Pythagorean triple:
936-1190-1514 calculated from 35² – 17², 2(35)(17), 35² + 17²

1513 is the Sum of Squares

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Today’s Puzzle:

How can you arrange 1513 dots into a perfect square when √1513 is irrational?

The answer is you arrange the dots into a centered square like this:

You can arrange them like that because 1513 is the sum of consecutive squares.

Factors of 1513:

  • 1513 is a composite number.
  • Prime factorization: 1513 = 17 × 89.
  • 1513 has no exponents greater than 1 in its prime factorization, so √1513 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 1513 has exactly 4 factors.
  • The factors of 1513 are outlined with their factor pair partners in the graphic below.

More about the Number 1513:

1513 is the sum of two squares in two different ways:
28² + 27²  = 1513, and
37² + 12²  = 1513.

1513 is the hypotenuse of FOUR Pythagorean triples:
55-1512-1513,  calculated from 28² – 27², 2(28)( 27), 28² + 27²
663-1360-1513, which is 17 times (39-80-89)
712-1335-1513, which is (8-15-17) times 89
888-1225-1513, calculated from 2(37)(12), 37² – 12², 37² + 12²

Could 1513 be a prime number?

Since its last two digits divided by 4 leave a remainder of 1, and 28² + 27² = 1513 with   28 and 27 having no common prime factors, 1513 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1513. Is 1513 divisible by 5, 13, 17, 29, or 37? Yes, it is divisible by 17, so 1513 is NOT a prime number.

37² + 12²  = 1513 and 37 and 12 have no common prime factors, so we could have arrived at the same result using those numbers.

Note: Numbers that are the sum of two squares in two or more ways are never prime.

 

1512 Bigger Bites of Cake

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Using the Cake Method:

I like using the cake method to find the prime factorization of a number. I also use it to find square roots.

If you know the multiplication table well, dividing by any number from 2 to 9 is not difficult.

It is easy to check to see if a number is divisible by 4 or by 9. And it is actually easier to divide by 4 once than it is to divide by 2 twice or to divide by 9 once than it is to divide by 3 twice. That way we get to take bigger bites of cake!

I know that 1512 is divisible by 4 because the number formed from the last two digits in order, 12, is divisible by 4.

I also know that 1512 is divisible by 9 because 1 + 5 + 1 + 2 = 9.

Thus, I’ll begin by dividing first by 4 and then the result by 9 as illustrated below:

When my new divisor becomes 42, if I didn’t remember where it appears in the multiplication table, I would still be fine. I know that 42 is divisible by 2 because it is even and 3 because 4 + 2 = 6, a number divisible by 3.

Thus I can divide 42 by 6 because 2 × 3 = 6, and it is far easier to divide 42 by 6 than it is to divide it first by 2 and then by 3.

To take the square root of 1512, I simply take the square root of the numbers on the outside of the cake. 4 and 9 are perfect squares so I use their square roots. Neither 6 nor 7 has any perfect squares, so I just multiply them together to get 42.

Dividing 1512 this way allowed me to make just a three-layer cake instead of a 6-layer cake to find its prime factorization and its square root!

Some people prefer to do yard work more than having cake, so here is one of the MANY possible factor trees for 1512. Make sure you pick up all seven of the leaves with prime numbers wherever they are.

Factors of 1512:

  • 1512 is a composite number.
  • Prime factorization: 1512 = 2 × 2 × 2 × 3 × 3 × 3 × 7, which can be written 1512 = 2³ × 3³ × 7.
  • 1512 has at least one exponent greater than 1 in its prime factorization so √1512 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1512 = (√36)(√42) = 6√42.
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (3 + 1)(3 + 1)(1 + 1) = 4 × 4 × 2 = 32. Therefore 1512 has exactly 32 factors.
  • The factors of 1512 are outlined with their factor pair partners in the graphic below.

 

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.