1635 The Logic to This Puzzle Is a Real Mystery

Today’s Puzzle:

You can easily find a way to start this puzzle, but just a few factors later, it’s a mystery what to do next. Give it a shot, and see what I mean.

Writing all the factor pairs for the clues often is helpful,

but not as helpful as we might hope this time. Here’s a video explaining what to do to find a few more factors of this puzzle using logic:

Factors of 1635:

  • 1635 is a composite number.
  • Prime factorization: 1635 = 3 × 5 × 109.
  • 1635 has no exponents greater than 1 in its prime factorization, so √1635 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 1635 has exactly 8 factors.
  • The factors of 1635 are outlined with their factor pair partners in the graphic below.

More about the Number 1635:

1635 is the hypotenuse of FOUR Pythagorean triples:
99-1632-1635, which is 3 times (33-544-545),
552-1539-1635, which is 3 times (184-513-545),
900-1365-1635, which is 15 times (60-91-109), and
981-1308-1635, which is (3-4-5) times 327.

OEIS.org informs us that there’s something special about the first nine decimals places of the fifth root of 1635.
Its fifth root is 4.392416875…
Can you figure out what is so special about that?

1634 Be Prepared for April Showers

Today’s Puzzle:

If you learn the multiplication and division facts in a standard multiplication table, you will be prepared to solve this somewhat tricky April Shower puzzle. You will also be able to solve MANY other mathematical challenges. Use logic to solve it, not guess and check, and it will be much less challenging to find the missing factors.

Factors of 1634:

  • 1634 is a composite number.
  • Prime factorization: 1634 = 2 × 19 × 43.
  • 1634 has no exponents greater than 1 in its prime factorization, so √1634 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 1634 has exactly 8 factors.
  • The factors of 1634 are outlined with their factor pair partners in the graphic below.

More about the Number 1634:

1634 is part of exactly two Pythagorean triples. Here are the formulas you can use to calculate those two triples:
2(817)(1), 817² – 1², 817² + 1, and
2(43)(19), 43² – 19², 43² + 19².

Do you see the factors of 1634 prominently displayed in those formulas?

1633 and Level 5

Today’s Puzzle:

It might be tricky in a few places, but use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues behave like a multiplication table.

Factors of 1633:

  • 1633 is a composite number.
  • Prime factorization: 1633 = 23 × 71.
  • 1633 has no exponents greater than 1 in its prime factorization, so √1633 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 1633 has exactly 4 factors.
  • The factors of 1633 are outlined with their factor pair partners in the graphic below.

More about the Number 1633:

1633 is the difference of two squares in two different ways:
817² – 816² = 1633, and
47² – 24² = 1633.

1631 The Importance of Practice

Today’s Puzzle:

I did not have the privilege of learning a musical instrument when I was growing up, but I did make sure my children had that opportunity. One of the topics discussed in this next episode of Bill Davidson’s Podcast is the importance that practice plays in both music and mathematics. I thought it was quite good.

I think practicing is best when it is enjoyable. If you solve this musical note puzzle, it will hopefully be an enjoyable way for you to practice a few multiplication and division facts. Just use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues will function like a multiplication table.

Factors of 1631:

  • 1631 is a composite number.
  • Prime factorization: 1631 = 7 × 233.
  • 1631 has no exponents greater than 1 in its prime factorization, so √1631 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 1631 has exactly 4 factors.
  • The factors of 1631 are outlined with their factor pair partners in the graphic below.

More about the Number 1631:

1631 is the hypotenuse of a Pythagorean triple:
735-1456-1631, which is 7 times (105-208-233).

1631 is the difference of two squares in two different ways:
816² – 815² = 1631, and
120² – 113² = 1631.

I found those number facts just from looking at the factors of 1631.

 

1630 and Level 3

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so those numbers and the given clues make the puzzle function like a multiplication table. Because this is a level 3 puzzle, first write the factors for 72 and 90. Then work your way down the puzzle row by row until you have found all the factors.

Factors of 1630:

  • 1630 is a composite number.
  • Prime factorization: 1630 = 2 × 5 × 163.
  • 1630 has no exponents greater than 1 in its prime factorization, so √1630 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 1630 has exactly 8 factors.
  • The factors of 1630 are outlined with their factor pair partners in the graphic below.

More about the Number 1630:

1630 is the hypotenuse of a Pythagorean triple:
978-1304-1630, which is (3-4-5) times 326.

1629 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work to make a multiplication table.

Factors of 1629:

1 + 6 + 2 = 9, so 1929 is divisible by both 3 and 9. (It’s only necessary to add the non-nine numbers together to check those two divisibility rules.)

  • 1629 is a composite number.
  • Prime factorization: 1629 = 3 × 3 × 181, which can be written 1629 = 3² × 181.
  • 1629 has at least one exponent greater than 1 in its prime factorization so √1629 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1629 = (√9)(√181) = 3√181.
  • 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 1629 has exactly 6 factors.
  • The factors of 1629 are outlined with their factor pair partners in the graphic below.

More about the Number 1629:

1629 is the sum of two squares:
30² + 27² = 1629.

1629 is the hypotenuse of a Pythagorean triple:
171-1620-1629, calculated from 30² – 27², 2(30)(27), 30² + 27².
It is also 9 times (19-180-181).

1628 A Simple Cross

Today’s Puzzle:

A simple cross is an appropriate symbol for Good Friday. Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1628:

  • 1628 is a composite number.
  • Prime factorization: 1628 = 2 × 2 × 11 × 37, which can be written 1628 = 2² × 11 × 37.
  • 1628 has at least one exponent greater than 1 in its prime factorization so √1628 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1628 = (√4)(√407) = 2√407.
  • 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 1628 has exactly 12 factors.
  • The factors of 1628 are outlined with their factor pair partners in the graphic below.

More about the Number 1628:

1628 is the hypotenuse of a Pythagorean triple:
528-1540-1628, which is (12-35-37) times 44.

1628 is the difference of two squares in two different ways:
408² – 406² = 1628, and
48² – 26²  = 1628.

1623 Easter Basket Challenge Puzzle

Today’s Puzzle:

Since I’ve recently made puzzles with a pink, purple, or blue Easter egg as well as some blades of grass blowing in the spring wind, it only makes sense that I would also give you an Easter basket in which to hold those other puzzles.

The puzzle is solved if you have written the numbers 1 to 10 in each of the boldly outlined areas of the puzzle, and if those numbers work with the clues to form four multiplication tables.

Print the puzzles or type the solution in this excel file: 12 Factors 1614-1623.

If you need a little help, here’s the same puzzle with the factor pairs for the clues written in.

And if you want even more help, here’s a 2 1/2 minute video on how to get started. I assume you already know the directions on how to solve this kind of puzzle that I gave at the top of this post.

Factors of 1623:

  • 1623 is a composite number.
  • Prime factorization: 1623 = 3 × 541.
  • 1623 has no exponents greater than 1 in its prime factorization, so √1623 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 1623 has exactly 4 factors.
  • The factors of 1623 are outlined with their factor pair partners in the graphic below.

More About the Number 1623:

1623 is the hypotenuse of a Pythagorean triple:
1023-1260-1623, which is 3 times (341-420-541).

1622 A Blue Egg for Your Easter Basket

Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution.

Math Eggs from Twitter:

Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

Factors of 1622:

  • 1622 is a composite number.
  • Prime factorization: 1622 = 2 × 811.
  • 1622 has no exponents greater than 1 in its prime factorization, so √1622 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 1622 has exactly 4 factors.
  • The factors of 1622 are outlined with their factor pair partners in the graphic below.

More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

1619 A Pink Egg Hidden in the Grass

Today’s Puzzle:

Easter is less than two weeks away. This pink puzzle is the first of three level-5 Easter eggs hidden amongst some blades of grass for you to find and solve. The puzzle might be a little tricky, but use logic every step of the way, and you’ll be able to find the unique solution:

Factors of 1619:

  • 1619 is a prime number.
  • Prime factorization: 1619 is prime.
  • 1619 has no exponents greater than 1 in its prime factorization, so √1619 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1619 has exactly 2 factors.
  • The factors of 1619 are outlined with their factor pair partners in the graphic below.

How do we know that 1619 is a prime number? If 1619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1619. Since 1619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1619 is a prime number.

More about the Number 1619:

1619 is the sum of two consecutive numbers:
809 + 810 = 1619.

1619 is also the difference of two consecutive squares:
810² – 809² = 1619.

What do you think about that?