1552 Look for Clues around the Corner

Today’s Puzzle:

This puzzle has four sets of clues that turn the corner. You will need to look around those corners to solve it. Use logic and have fun!

Factors of 1552:

  • 1552 is a composite number.
  • Prime factorization: 1552 = 2 × 2 × 2 × 2 × 97, which can be written 1552 = 2⁴ × 97
  • 1552 has at least one exponent greater than 1 in its prime factorization so √1552 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1552 = (√16)(√97) = 4√97
  • 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 1552 has exactly 10 factors.
  • The factors of 1552 are outlined with their factor pair partners in the graphic below.

More about the Number 1552:

1552 is the sum of two squares:
36² + 16²  = 1552

1552 is the hypotenuse of a Pythagorean triple:
1040-1152-1552, calculated from 36² – 16², 2(36)(16), 36² + 16².
It is also 16 times (65-72-97).

OEIS.org looked around the corner at the two numbers preceding 1552 to find something special about that number: The sum of its prime factors equals the sum of the prime factors of those previous two numbers! That’s a cool enough fact that I decided to make this graphic:

1530 Jack-o’-lantern

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.

 

 

1517 and Level 5

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.

1506 and Level 5

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.

1494 and Level 5

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of those factors. Be sure to use logic to find the factors! Guessing and checking will only frustrate you.

Factors of 1494:

  • 1494 is a composite number.
  • Prime factorization: 1494 = 2 × 3 × 3 × 83, which can be written 1494 = 2 × 3² × 83
  • 1494 has at least one exponent greater than 1 in its prime factorization so √1494 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1494 = (√9)(√166) = 3√166
  • 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 1494 has exactly 12 factors.
  • The factors of 1494 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1494:

1494 is not the sum of or the difference of two squares, but it is still a part of three Pythagorean triples because of these three ways it can be factored:
1494 = 2(747)(1),
1494 = 2(249)(3), and
1494 = 2(83)(9).
And because for whole numbers where a > b, 2(a)(b), a² – b², a² + b² will be a Pythagorean triple.

1483 and Level 5

Today’s Puzzle:

Some of the clues in the same row or column in this puzzle have more than one common factor. In each case, will you make the logical choice to find the puzzle’s unique solution?

Factors of 1483:

  • 1483 is a prime number.
  • Prime factorization: 1483 is prime.
  • 1483 has no exponents greater than 1 in its prime factorization, so √1483 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1483 has exactly 2 factors.
  • The factors of 1483 are outlined with their factor pair partners in the graphic below.

How do we know that 1483 is a prime number? If 1483 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1483. Since 1483 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1483 is a prime number.

More about the Number 1483:

1483 is the difference of two squares:
742² – 741² = 1483

The first five prime decades are listed below. 1483 is the second prime number in the fifth prime decade:

1473 A Level 5 Designed by My Granddaughter

Today’s Puzzle:

My son and his family arrived today for a two-month long visit. We are so excited that they are here. His nine-year-old daughter has solved her share of Find the Factors puzzles. Today I gave her the opportunity to help me design one which she thoroughly loved doing. Here is the first of her creations. Will you be able to solve it? It’s a level 5 so you might find it a bit tricky.

Factors of 1473:

  • 1473 is a composite number.
  • Prime factorization: 1473 = 3 × 491
  • 1473 has no exponents greater than 1 in its prime factorization, so √1473 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 1473 has exactly 4 factors.
  • The factors of 1473 are outlined with their factor pair partners in the graphic below.

More about the number 1473:

1473 can be written as the difference of two squares in two different ways:
737² – 736² = 1473
247² – 244² = 1473

1460 and Level 5

Today’s Puzzle:

Can you use logic to figure out where the numbers from 1 to 10 must go so that the given clues and the factors you find make this puzzle a multiplication table?

Factors of 1460:

  • 1460 is a composite number.
  • Prime factorization: 1460 = 2 × 2 × 5 × 73, which can be written 1460 = 2² × 5 × 73
  • 1460 has at least one exponent greater than 1 in its prime factorization so √1460 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1460 = (√4)(√365) = 2√365
  • 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 1460 has exactly 12 factors.
  • The factors of 1460 are outlined with their factor pair partners in the graphic below.

Other Facts about 1460:

As I mentioned earlier, √1460 = 2√365.
Most years have 365 days in them, but not 2020, our current year. Nevertheless,
1460 ÷ 73 × 101= 2020.

1460 is the sum of two squares in two different ways:
28² + 26² = 1460
38² + 4² =1460

1460 is the hypotenuse of FOUR Pythagorean triples:
108-1456-1460, calculated from 28² – 26², 2(28)(26), 28² + 26²
304-1428-1460, calculated from 2(38)(4), 38² – 4², 38² + 4²
876-1168-1460 which is (3-4-5) times 292
960-1100-1460 which is 20 times (48-55-73)

1447 Christmas Light Puzzle

If you’ve ever had a string of lights go out because ONE bulb went bad, it can be a very frustrating puzzle to figure out which light is causing the problem.

This is not that kind of puzzle. For this one, you just need to figure out where to put the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of those numbers. There is only one solution, and if you always use logic, it will not be a frustrating puzzle to solve.

I gave that puzzle the puzzle number 1447. That number won’t help you solve the puzzle, but here are some facts about it anyway:

  • 1447 is a prime number.
  • Prime factorization: 1447 is prime.
  • 1447 has no exponents greater than 1 in its prime factorization, so √1447 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1447 has exactly 2 factors.
  • The factors of 1447 are outlined with their factor pair partners in the graphic below.

How do we know that 1447 is a prime number? If 1447 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1447. Since 1447 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1447 is a prime number.

1447 is also the difference of two consecutive squares:
724² – 723² = 1447

 

 

 

1438 and Level 5

You can solve this puzzle by using logic and multiplication/division facts. The unique solution requires all the numbers from 1 to 10 in both the first column and the top row. Can you solve it?

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

Today’s puzzle was given the number 1438. Here are a few facts about that number:

  • 1438 is a composite number.
  • Prime factorization: 1438 = 2 × 719
  • 1438 has no exponents greater than 1 in its prime factorization, so √1438 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 1438 has exactly 4 factors.
  • The factors of 1438 are outlined with their factor pair partners in the graphic below.

1438 is the sum of four consecutive numbers:
358  + 359 + 360 + 361 = 1438