1601 and Level 6

Today’s Puzzle:

Remember to use logic for EVERY step when solving this puzzle. Guessing and checking will likely just frustrate you! It’s a level 6 puzzle, so it could be tricky.
Keep in mind:
1 and 2 are common factors of 6 and 8,
3 and 9 are common factors of 9 and 27,
4 & 8 are common factors of 32 and 8, and
4, 5, & 10 are common factors of 20 & 40.

As always, there is only one solution.

Factors of 1601:

  • 1601 is a prime number.
  • Prime factorization: 1601 is prime.
  • 1601 has no exponents greater than 1 in its prime factorization, so √1601 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1601 has exactly 2 factors.
  • The factors of 1601 are outlined with their factor pair partners in the graphic below.

How do we know that 1601 is a prime number? If 1601 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1601. Since 1601 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1601 is a prime number.

More about the Number 1601:

1601 is one more than a perfect square, so it is the sum of two squares:
40² + 1² = 1601.

1601 is the hypotenuse of a Pythagorean triple:
80-1599-1601, calculated from 2(40)(1), 40² – 1², 40² + 1².

Here’s another way we know that 1601 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 1² = 1601 with 40 and 1 having no common prime factors, 1601 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1601. Since 1601 is not divisible by 5, 13, 17, 29, or 37, we know that 1601 is a prime number.

1591 Conversation Heart

Today’s Puzzle:

If this valentine-shaped level 6 puzzle gets kids talking about multiplication, then it will truly be a conversation heart.

Factors of 1591:

  • 1591 is a composite number.
  • Prime factorization: 1591 = 37 × 43.
  • 1591 has no exponents greater than 1 in its prime factorization, so √1591 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 1591 has exactly 4 factors.
  • The factors of 1591 are outlined with their factor pair partners in the graphic below.

More about the Number 1591:

1591 is the difference of two squares two different ways:
796² – 795² = 1591, and
40² – 3² = 1591. That means we are only 3², or 9 numbers away from the next perfect square, 40², or 1600.

1582 Half the Time You’ll Be Thinking about Multiples instead of Factors

Today’s Puzzle:

In the level 6 puzzle, the possible common factors of 40 and 20 are 4, 5, and 10 while the possible common factors of 24 and 12 are 3, 4, and 6.

Don’t worry about which common factor to choose to start the puzzle. Instead, think about multiples. One of the numbers from 1 to 10 will only be able to go in one place in the top row because that’s the only place one of its multiples is in the column below it. That same number will also have only one place it can go in the first column.

You will likely look for lone multiples of particular numbers a total of five times as you solve this puzzle. Good luck!

Factors of 1582:

  • 1582 is a composite number.
  • Prime factorization: 1582 = 2 × 7 × 113.
  • 1582 has no exponents greater than 1 in its prime factorization, so √1582 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 1582 has exactly 8 factors.
  • The factors of 1582 are outlined with their factor pair partners in the graphic below.

More about the Number 1582:

1582 is the hypotenuse of a Pythagorean triple:
210-1568-1582, which is 14 times (15-112-113).

1582 is the sum of the four numbers from 394 to 397.

1582 is the sum of the seven numbers from 223 to 229.

1582 is the sum of the twenty-eight numbers from 43 to 70.

 

1564 Two Candles

Today’s Puzzle:

Candles that are lit in the darkness can be seen from quite a distance away.  Candles and candlelight are symbols of Christmas. The babe born on that first Christmas day would become the Light of the World.

What makes a level 6 puzzle more difficult? Can you see that the common factor of 60 and 30 might be 5, 6, or 10? Which one should you use? The other two won’t work with the other clues in the puzzle.

Likewise, the common factor of 48 and 12 might be 4, 6, or 12. Don’t guess which one to use! Use logic, and find the solution to this puzzle.

One blank row and one blank column intersect in a single cell. Can you determine what number belongs in that cell before you write any other factors? That is the first thing I would do.

Here is the same puzzle without any added color:

Factor Tree for 1564:

64 is divisible by 4, so 1564 is also. Here is a factor tree for 1564 that divisibility fact:

Factors of 1564:

  • 1564 is a composite number.
  • Prime factorization: 1564 = 2 × 2 × 17 × 23, which can be written 1564 = 2² × 17 ×
  • 1564 has at least one exponent greater than 1 in its prime factorization so √1564 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1564 = (√4)(√391) = 2√
  • 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 1564 has exactly 12 factors.
  • The factors of 1564 are outlined with their factor pair partners in the graphic below.

 

More about the Number 1564:

1564 is the hypotenuse of a Pythagorean triple:
736-1380-1564, which is (8-15-17) times 92.

1564 is the difference of two squares in two different ways:
392² – 390² = 1564, and
40² – 6² = 1564. That means we are only 36 numbers away from 40² = 1600.

1564 is in this cool pattern:

1553 Ornamental Corn

Today’s Puzzle:

Ornamental corn is a popular decoration at Thanksgiving. Today’s puzzle looks a little bit like ornamental corn, and there’s at least a kernel of truth to that statement! Solve the puzzle, and I will think YOU are a-maize-ing!

Here’s the same puzzle if you want to print it in black and white:

Factors of 1553:

  • 1553 is a prime number.
  • Prime factorization: 1553 is prime.
  • 1553 has no exponents greater than 1 in its prime factorization, so √1553 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1553 has exactly 2 factors.
  • The factors of 1553 are outlined with their factor pair partners in the graphic below.

How do we know that 1553 is a prime number? If 1553 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1553. Since 1553 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1553 is a prime number.

More About the Number 1553:

1553 is the sum of the squares of two numbers that are reverses of each other:
32² + 23² = 1553

1553 is the hypotenuse of a primitive Pythagorean triple:
495-1472-1553, calculated from 32² – 23², 2(32)(23), 32² + 23².

Here’s another way we know that 1553 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 23² = 1553 with 32 and 23 having no common prime factors, 1553 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1553. Since 1553 is not divisible by 5, 13, 17, 29, or 37, we know that 1553 is a prime number.

1532 Don’t Let This Puzzle Spook You!

Today’s Puzzle:

If this little ghost prowls your neighborhood this Halloween, don’t let it spook you. Sure, it is a level 6 puzzle, but if you stick to using logic from start to finish, you’ll know the most about this ghost!

Here’s the same puzzle minus the embellishments:

If you know what a normal distribution is, then you should enjoy this statistics joke:

Factors of 1532:

  • 1532 is a composite number.
  • Prime factorization: 1532 = 2 × 2 × 383, which can be written 1532 = 2² × 383
  • 1532 has at least one exponent greater than 1 in its prime factorization so √1532 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1532 = (√4)(√383) = 2√383
  • 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 1532 has exactly 6 factors.
  • The factors of 1532 are outlined with their factor pair partners in the graphic below.

More about the Number 1532:

1532 = 2 × 383 × 2. That factorization looks the same frontwards or backward.

1532 can be written as the difference of two squares:
384² – 382² = 1532.

 

 

1518 and Level 6

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

1507 and Level 6

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).

1495 and Level 6

Today’s Puzzle:

Hint: The only way 12 can be put in the first column of this puzzle is to let one of the 60’s use it. We don’t have to know which 60 that is, to know that a 5 will go above that 60 in the top row. Knowing that, where does 5 have to go in the first column? That’s the logic needed to get started on this puzzle:

Factors of 1495:

  • 1495 is a composite number.
  • Prime factorization: 1495 = 5 × 13 × 23.
  • 1495 has no exponents greater than 1 in its prime factorization, so √1495 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 1495 has exactly 8 factors.
  • The factors of 1495 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1495:

1495 is the hypotenuse of FOUR Pythagorean triples:
368-1449-1495, which is 23 times (16-63-65)
575-1380-1495, which is (5-12-13) times 115,
759-1288-1495, which is 23 times (33-56-65), and
897-1196-1495, which is (3-4-5) times 299.

 

 

1484 A Popsicle for a Hot Summer Day

Today’s Puzzle:

It can be tricky to eat a melting popsicle on a hot summer day. Likewise, it can be tricky to solve a level 6 puzzle even if it looks like a popsicle!  Making it was my granddaughter’s idea. We hope you enjoy it!

Factors of 1484:

If you know that 7 x 12 is 84, it isn’t hard to recognize that 1484 is divisible by 7.

  • 1484 is a composite number.
  • Prime factorization: 1484 = 2 × 2 × 7 × 53, which can be written 1484 = 2² × 7 × 53
  • 1484 has at least one exponent greater than 1 in its prime factorization so √1484 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1484 = (√4)(√371) = 2√371
  • 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 1484 has exactly 12 factors.
  • The factors of 1484 are outlined with their factor pair partners in the graphic below.

Other facts about the Number 1484:

1484 is the difference of two squares in two different ways:
372² – 370² = 1484
60² – 46² = 1484

784-1260-1484 which is 28 times (28-45-53)