1635 The Logic to This Puzzle Is a Real Mystery

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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

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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

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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.

Math Happens When Two of 1632’s Factors Look in a Mirror!

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Today’s Puzzle:

Both 12 and 102 are factors of 1632. Something special happens when either one squares itself and looks in a mirror. Solving this puzzle from Math Happens will show you what happens to 12 and 12².

You can see that puzzle on page 33 of this e-edition or this pdf of the Austin Chronicle. You can find other Math Happens Puzzles here.

This next puzzle will help you discover what happens when 102 and 102² look in a mirror!

Why do you suppose the squares of (12, 21) and (102, 201) have that mirror-like property?

Factor Trees for 1632:

There are many possible factor trees for 1632, but today I will focus on two trees that use factor pairs containing either 12 or 102:

Factors of 1632:

  • 1632 is a composite number.
  • Prime factorization: 1632 = 2 × 2 × 2 × 2 × 2 × 3 × 17, which can be written 1632 = 2⁵ × 3 × 17.
  • 1632 has at least one exponent greater than 1 in its prime factorization so √1632 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1632 = (√16)(√102) = 4√102.
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1632 has exactly 24 factors.
  • The factors of 1632 are outlined with their factor pair partners in the graphic below.

More about the Number 1632:

1632 is the hypotenuse of a Pythagorean triple:
768-1440-1632, which is (8-15-17) times 96.

1632 is the difference of two squares in EIGHT different ways:
409² – 407² = 1632,
206² – 202² = 1632,
139² – 133² = 1632,
106² – 98² = 1632,
74² – 62² = 1632,
59² – 43² = 1632,
46² – 22² = 1632, and
41² – 7² = 1632.

That last difference of two squares means 1632 is only 49 numbers away from the next perfect square, 1681.

 

1631 The Importance of Practice

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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

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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

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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

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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.

1627 Color-Coded Prime Numbers

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Today’s Puzzle:

Study this color-coded chart of prime numbers. 1627 is the smallest prime number that begins something special. Can you figure out what that is?

Also, why do you think I’ve underlined some of the other prime numbers on the list?

Memorizing which numbers are prime can be a big time-saver in mathematics. How many in a row can you recite without looking?

Factors of 1627:

  • 1627 is a prime number.
  • Prime factorization: 1627 is prime.
  • 1627 has no exponents greater than 1 in its prime factorization, so √1627 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1627 has exactly 2 factors.
  • The factors of 1627 are outlined with their factor pair partners in the graphic below.

How do we know that 1627 is a prime number? If 1627 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1627. Since 1627 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1627 is a prime number.

More about the Number 1627:

1627 is the sum of two consecutive numbers:
813 + 814 = 1627.

1627 is also the difference of two consecutive squares:
814² – 813² = 1627.

What do you think of that?

1626 is a Centered Pentagonal Number

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Today’s Puzzle:

1626 is the 26th centered pentagonal number and it is also one more than 5 times the 25th triangular number.

Today’s puzzle is for you to figure out why the following two expressions are equivalent. The first expression is the formula for the nth centered pentagonal number, and the second expression is one more than 5 times the formula for the nth triangular number.
(5n²-5n+2)/2, when n = 26, and
1 + 5n(n+1)/2, when n = 25.

Factors of 1626:

  • 1626 is a composite number.
  • Prime factorization: 1626 = 2 × 3 × 271.
  • 1626 has no exponents greater than 1 in its prime factorization, so √1626 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 1626 has exactly 8 factors.
  • The factors of 1626 are outlined with their factor pair partners in the graphic below.

More about the Number 1626:

2(813)(1) = 1626 means that 2(813)(1), 813² – 1², 813² + 1² is a Pythagorean triple, and
2(271)(3) = 1626 means that 2(271)(3), 271² – 3², 271² + 3² is as well.

Calculate those expressions and you will have found the only two Pythagorean triples containing the number 1626.