1535 Your Favorite Candy Bar

Today’s Puzzle:

If you could have your favorite candy bar in the largest size possible, would it actually turn out to be more than you can chew? Of course not. Even if it took a long time for you to finish that candy bar, it would not be an impossible challenge.

Think of this Challenge puzzle the same way. I think this one uses some delightful logic. Find that logic, take your time, and enjoy the process!

Print the puzzles or type the solution in this excel file: 10 Factors 1526-1535

Factors of 1535:

  • 1535 is a composite number.
  • Prime factorization: 1535 = 5 × 307.
  • 1535 has no exponents greater than 1 in its prime factorization, so √1535 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 1535 has exactly 4 factors.
  • The factors of 1535 are outlined with their factor pair partners in the graphic below.

Another Fact about the number 1535:

1535 is the hypotenuse of a Pythagorean triple:
921-1228-1535 which is (3-4-5) times 307.

1510 Challenge Puzzle

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.

 

1499 Challenge Puzzle

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.

 

 

1487 A Challenging Puzzle

Today’s Puzzle:

This puzzle is like four out-of-order 1 – 10 multiplication tables that work together. The clues in the puzzle need to be the products of the numbers you write. It won’t be easy, but use logic to solve it. Good luck!

Print the puzzles or type the solution in this excel file: 10 Factors 1478-1487

Factors of 1487:

  • 1487 is a prime number.
  • Prime factorization: 1487 is prime.
  • 1487 has no exponents greater than 1 in its prime factorization, so √1487 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1487 has exactly 2 factors.
  • The factors of 1487 are outlined with their factor pair partners in the graphic below.

How do we know that 1487 is a prime number? If 1487 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1487. Since 1487 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1487 is a prime number.

Other Facts about the Number 1487:

1487 is the difference of two squares:
744²-743² = 1487.

1487 is the third prime number in the fifth prime decade. See it in the list below:

1477 Challenge

Today’s Puzzle:

It’s been a few days since I’ve last looked at this puzzle, but I was still able to solve it in 17 minutes. Can you beat my time?

Print the puzzles or type the solution in this excel file: 12 Factors 1468-1477

Factors of 1477:

  • 1477 is a composite number.
  • Prime factorization: 1477 = 7 × 211
  • 1477 has no exponents greater than 1 in its prime factorization, so √1477 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 1477 has exactly 4 factors.
  • The factors of 1477 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1477:

1477 is the difference of two squares in two different ways:
739² – 738² = 1477
109² – 102² = 1477

1467 Challenge Puzzle

Challenge Puzzle:

Use logic to solve this Challenge Puzzle. The given clues work together to make finding the unique solution a little easier than usual. Have fun with it!

Print the puzzles or type the solution in this excel file: 10 Factors 1454-1467

Factors of 1467:

1467 is divisible by 3 because 1 + 4 = 5 and 5, 6, 7 are three consecutive numbers. Since the middle number, 6, is divisible by 3, we know that 1467 is also divisible by 9.

Of course, you could also add up the digits of 1467 to get 1 + 4 + 6 + 7 = 27, a number divisible by both 3 and 9, to know that 1467 is divisible by both 3 and 9.

  • 1467 is a composite number.
  • Prime factorization: 1467 = 3 × 3 × 163, which can be written 1467 = 3² × 163
  • 1467 has at least one exponent greater than 1 in its prime factorization so √1467 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1467 = (√9)(√163) = 3√163
  • 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 1467 has exactly 6 factors.
  • The factors of 1467 are outlined with their factor pair partners in the graphic below.

 

 

1429 Find the Factors Challenge

I wanted today’s puzzle to look like a big candy bar, but I don’t think I succeeded. I hope you will still think it is the best treat you got today! Good luck!

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

Now I’ll tell you some facts about the number 1429:

  • 1429 is a prime number.
  • Prime factorization: 1429 is prime.
  • 1429 has no exponents greater than 1 in its prime factorization, so √1429 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1429 has exactly 2 factors.
  • The factors of 1429 are outlined with their factor pair partners in the graphic below.

How do we know that 1429 is a prime number? If 1429 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1429. Since 1429 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1429 is a prime number.

OEIS.org informs us that 1429² = 2,042,041. That’s the smallest perfect square whose first three digits are repeated in order by the next three digits.

1429 is the sum of two squares:
30² + 23² = 1429

1429 is the hypotenuse of a primitive Pythagorean triple:
371-1380-1429 calculated from 30² – 23², 2(30)(23), 30² + 23²

Here’s another way we know that 1429 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 23² = 1429 with 30 and 23 having no common prime factors, 1429 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1429. Since 1429 is not divisible by 5, 13, 17, 29, or 37, we know that 1429 is a prime number.

 

1418 Challenge Puzzle

The 19 clues in this Find the Factors Challenge Puzzle are enough to find its unique solution. Can you find it?

Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll write a few facts about the puzzle number, 1418:

  • 1418 is a composite number.
  • Prime factorization: 1418 = 2 × 709
  • 1418 has no exponents greater than 1 in its prime factorization, so √1418 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 1418 has exactly 4 factors.
  • The factors of 1418 are outlined with their factor pair partners in the graphic below.

1418 is the sum of two squares:
37² + 7² = 1418

518-1320-1418 calculated from 2(37)(7), 37² – 7², 37² + 7².
It is also 2 times (259-660-709)

 

1403 Multiplication Table Challenge

Just because you’re not in elementary school anymore doesn’t mean that the multiplication table can’t be a challenge. This one certainly is. Can you write the numbers 1 to 10 in the four factor areas so that this multiplication table works with the given clues? Don’t get discouraged; it will probably take you at least 15 minutes just to put those factors in the right places.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll share some information about the puzzle number, 1403:

  • 1403 is a composite number.
  • Prime factorization: 1403 = 23 × 61
  • 1403 has no exponents greater than 1 in its prime factorization, so √1403 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 1403 has exactly 4 factors.
  • The factors of 1403 are outlined with their factor pair partners in the graphic below.

1403 is the hypotenuse of a Pythagorean triple:
253-1380-1403 which is 23 times (11-60-61)

 

 

1377 Easter Basket Challenge

Occasionally,  we hear that the number of Easter eggs that are found is one or two less than the number of eggs that were hidden. Still most of the time, all the eggs and candies do get found. You really have no trouble finding all those goodies, and the Easter Egg Hunt seems like it is over in seconds.  You can find Easter Eggs but can you find factors? Here’s an Easter Basket Find the Factors 1 – 10 Challenge Puzzle for you. I guarantee it won’t be done in seconds. Can you find all the factors? I dare you to try!

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll mention a few facts about the number 1377:

  • 1377 is a composite number.
  • Prime factorization: 1377 = 3 × 3 × 3 × 3 × 17, which can be written 1377 = 3⁴ × 17
  • 1377 has at least one exponent greater than 1 in its prime factorization so √1377 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1377 = (√81)(√17) = 9√17
  • 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 1377 has exactly 10 factors.
  • The factors of 1377 are outlined with their factor pair partners in the graphic below.

1377 is the sum of two squares:
36² + 9² = 1377

1377 is the hypotenuse of a Pythagorean triple:
648-1215-1377 which is (8-15-17) times 81
and can also be calculated from 2(36)(9), 36² – 9², 36² + 9²