Puzzles

1640 A Level 2 Flower

/ ivasallay / Leave a comment

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1640:

  • 1640 is a composite number.
  • Prime factorization: 1640 = 2 × 2 × 2 × 5 × 41, which can be written 1640 = 2³ × 5 × 41.
  • 1640 has at least one exponent greater than 1 in its prime factorization so √1640 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1640 = (√4)(√410) = 2√410.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1640 has exactly 16 factors.
  • The factors of 1640 are outlined with their factor pair partners in the graphic below.

More about the Number 1640:

Since 1640 = 40 x 41, we can be sure that 1640 is the sum of the first 40 even numbers.

1640 is the sum of two squares in two different ways:
38² + 14² = 1640, and
34² + 22² = 1640.

1640 is the hypotenuse of a Pythagorean triple in FOUR different ways:
360-1600-1640, which is 40 times (9-40-41),
672-1496-1640, calculated from 34² – 22², 2(34)(22), 34² + 22²,
but it is also 8 times (84-187-205),
984-1312-1640, which is (3-4-5) times 328, and
1064-1248-1640, calculated from 2(38)(14), 38² – 14², 38² + 14²,
but it is also 8 times (133-156-205).

1640₁₀ = 2222₉ because 2(9³ + 9² + 9¹ + 9⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.
1640₁₀ = 2020202₃ because 2(3⁶ + 3⁴ + 3² + 3⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.

 

1639 and Level 1

/ ivasallay / Leave a comment

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will make this puzzle function like a multiplication table.

Factors of 1639:

1 – 6 + 3 – 9 = -11 so 1639 is divisible by 11.

  • 1639 is a composite number.
  • Prime factorization: 1639 = 11 × 149.
  • 1639 has no exponents greater than 1 in its prime factorization, so √1639 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 1639 has exactly 4 factors.
  • The factors of 1639 are outlined with their factor pair partners in the graphic below.

More about the Number 1639:

1639 is the hypotenuse of a Pythagorean triple:
561 1540 1639, which is 11 times (51-140-149).

1639 is the 22nd nonagonal number because
22(7·22 – 5)/2 =
22(154 – 5)/2=
22(149)/2 =
11(149) = 1639.
Mathworld.Wolfram has illustrations of the first 5 nonagonal numbers.

1637 Flower Challenge

/ ivasallay / Leave a comment

Today’s Puzzle:

Mother’s Day in the United States is this Sunday so I made this challenging flower puzzle for the occasion.

Use logic to write the numbers from 1 to 10 in each of the four boldly outlined areas so that those numbers and the given clues work together to make four multiplication tables.

Factors of 1637:

  • 1637 is a prime number.
  • Prime factorization: 1637 is prime.
  • 1637 has no exponents greater than 1 in its prime factorization, so √1637 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1637 has exactly 2 factors.
  • The factors of 1637 are outlined with their factor pair partners in the graphic below.

How do we know that 1637 is a prime number? If 1637 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1637. Since 1637 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1637 is a prime number.

More about the Number 1637:

1637 is the sum of two squares:
31² + 26² = 1637.

1637 is the hypotenuse of a Pythagorean triple:
285-1612-1637, calculated from 31² – 26², 2(31)(26), 31² + 26².

Here’s another way we know that 1637 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 26² = 1637 with 31 and 26 having no common prime factors, 1637 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1637. Since 1637 is not divisible by 5, 13, 17, 29, or 37, we know that 1637 is a prime number.

Do you notice anything else special about the number 1637 in this color-coded chart?

1636 A Tricky Mystery Puzzle

/ ivasallay / Leave a comment

Today’s Puzzle:

It took me a while to figure out the logic of this puzzle so don’t get discouraged if you find it a bit tricky, too.

If its logic eludes you, this short video I posted on Twitter should help.

Factors of 1636:

  • 1636 is a composite number.
  • Prime factorization: 1636 = 2 × 2 × 409, which can be written 1636 = 2² × 409.
  • 1636 has at least one exponent greater than 1 in its prime factorization so √1636 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1636 = (√4)(√409) = 2√409.
  • 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 1636 has exactly 6 factors.
  • The factors of 1636 are outlined with their factor pair partners in the graphic below.

More about the Number 1636:

1636 is the sum of two squares:
40² + 6² = 1636.

1636 is the hypotenuse of a Pythagorean triple:
480-1564-1636, calculated from 2(40)(6), 40² – 6², 40² + 6².
It is also 4 times (120-391-409).

1636⁴ = 7,163,630,838,016. Thank you OEIS.org for that fun fact!

1635 The Logic to This Puzzle Is a Real Mystery

/ ivasallay / 4 Comments

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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).