1674 and Level 2

Today’s Puzzle:

Make a multiplication table out of this puzzle. Can you see how to do it? The factors won’t be in the usual order, but I’m sure you can figure it out!

Factors of 1674:

  • 1674 is a composite number.
  • Prime factorization: 1674 = 2 × 3 × 3 × 3 × 31, which can be written 1674 = 2 × 3³ × 31.
  • 1674 has at least one exponent greater than 1 in its prime factorization so √1674 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1674 = (√9)(√186) = 3√186.
  • The exponents in the prime factorization are 1,3 and 1. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1674 has exactly 16 factors.
  • The factors of 1674 are outlined with their factor pair partners in the graphic below.

More about the number 1674:

From OEIS.org we learn that
1/1 + 1/2 + 1/3 + 1/4 + . . . + 1/1672 + 1/1673 ≈ 7.999888, but if you add the next tiny fraction, 1/1674, the sum will be a tiny bit more than 8 or approximately 8.000486.

That’s adding a whole lot of unit fractions just to get a sum over 8.

 

1664 and Level 2

Today’s Puzzle:

Write the numbers 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.

Factor Cake for 1664:

We can make a factor cake for 1664 by doing some successive divisions. Divide 1664 by 2, divide that answer by 2, and so forth until you make a factor cake that looks like this:

Factors of 1664:

  • 1664 is a composite number.
  • Prime factorization: 1664 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 13, which can be written 1664 = 2⁷ × 13.
  • 1664 has at least one exponent greater than 1 in its prime factorization so √1664 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1664 = (√64)(√26) = 8√26.
  • The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1664 has exactly 16 factors.
  • The factors of 1664 are outlined with their factor pair partners in the graphic below.

More About the Number 1664:

1664 is the sum of two squares:
40² + 8² = 1664.
That happened because it has a prime factor that leaves a remainder of 1 when divided by 4 AND all of its other prime factors are powers of 2 or perfect squares:

But that’s not all that cool about 1664. What patterns do you notice below?
2(24² + 16²) = 1664,
4(20² + 4²) = 1664,
8(12² + 8²) = 1664,
16(10² + 2²) = 1664,
32(6² + 4²) = 1664,
64(5² + 1²) = 1664, and
128(3² + 2²) = 1664.

1664 is the hypotenuse of a Pythagorean triple:
640-1536-1664, calculated from 2(40)(8), 40² – 8², 40² + 8².
That triple is also (5-12-13) times 128.

1651 Multiplication Fun

Today’s Puzzle:

Look how much fun these kids are having doing multiplication!

A game like that can help kids get ready to solve a fun puzzle based on the multiplication table.

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

Factors of 1651:

  • 1651 is a composite number.
  • Prime factorization: 1651 = 13 × 127.
  • 1651 has no exponents greater than 1 in its prime factorization, so √1651 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 1651 has exactly 4 factors.
  • The factors of 1651 are outlined with their factor pair partners in the graphic below.

More About the Number 1651:

1651 is the hypotenuse of a Pythagorean triple:
635-1524-1651, which is (5-12-13) times 127.

1651 is the 26th heptagonal number because
5(26²)/2 – 3(26)/2 = 1651.

1651 is a nice-looking palindrome in base 2:
1651₁₀ = 11001110011₂.
That just means that
2¹⁰ + 2⁹ + 2⁶ + 2⁵ + 2⁴+ 2¹+ 2⁰ = 1024 + 512 + 64 + 32 + 16 + 2 + 1 = 1651.

 

 

1640 A Level 2 Flower

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.

 

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).

1615 Should Today Be Root Ten Day?

Today’s Puzzle:

In a recent post, I compared π or (3.14…) with √10 or (3.16…). Steve Morris lives in England where today’s date is written 16-3, not 3-16. He jokingly commented, “So I guess Tuesday (16 March) should be Root Ten Day!” Seriously, day-month-year makes more sense as a writing convention than month-day-year.

Should today be Root Ten Day?
14 March has long been embraced as pi day in the United States, but should 16 March also be a quasi-holiday where kids eat roots like ten French fries or ten carrot sticks?

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentines day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

To which we could add
√1 is 1, and New Year’s Day is January 1, and
√10 is about 3.1, and Halloween is October 31. (I realize there is a rounding issue with that one.)

Oops. That could be said about all the fake holidays I’ve listed above.

And here’s a more serious thought:

Well, however you want to remember what √10 is or not, I decided to make today’s puzzle look like a square root sign for the fun of it. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1615:

  • 1615 is a composite number.
  • Prime factorization: 1615 = 5 × 17 × 19.
  • 1615 has no exponents greater than 1 in its prime factorization, so √1615 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 1615 has exactly 8 factors.
  • The factors of 1615 are outlined with their factor pair partners in the graphic below.

More about the Number 1615:

1615 is the hypotenuse of FOUR Pythagorean triples:
247-1596-1615, which is 19 times (13-84-85),
684-1463-1615, which is 19 times (36-77-85),
760-1425-1615, which is (8-15-17) times 95, and
969-1292-1615, which is (3-4-5) times 323.

1606 A Lucky Shamrock

Today’s Puzzle:

Even if you don’t know all the factors of some of these clues in this shamrock puzzle, there are enough that you will know, and you can figure the rest out easily. Lucky you!

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1606:

  • 1606 is a composite number.
  • Prime factorization: 1606 = 2 × 11 × 73.
  • 1606 has no exponents greater than 1 in its prime factorization, so √1606 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 1606 has exactly 8 factors.
  • The factors of 1606 are outlined with their factor pair partners in the graphic below.

More about the Number 1606:

1606 is the hypotenuse of a Pythagorean triple:
1056-1210-1606. which is 22 times (48-55-73).

1606 is in a couple of other Pythagorean triples that can be calculated from
2(803)(1), 803² – 1², 803² + 1², and
2(73)(11), 73² – 11², 73² + 11².

1595 and Level 2

Today’s Puzzle:

This puzzle is just a multiplication table whose missing factors are not in the usual order. Can you figure out where the factors from 1 to 10 should go?

Factors of 1595:

1595 ends with a 5, so it is divisible by 5.
1 – 5 + 9 – 5 = 0, so 1595 is divisible by 11.

  • 1595 is a composite number.
  • Prime factorization: 1595 = 5 × 11 × 29.
  • 1595 has no exponents greater than 1 in its prime factorization, so √1595 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 1595 has exactly 8 factors.
  • The factors of 1595 are outlined with their factor pair partners in the graphic below.

More about the Number 1595:

1595 is the hypotenuse of FOUR Pythagorean triples:
187-1584-1595, which is 11 times (17-144-145),
264-1573-1595, which is 11 times (24-143-145),
957-1276-1595, which is (3-4-5) times 319, and
1100-1155-1595, which is (20-21-29) times 55.

1587 XOXOXO Hugs and Kisses OXOXOX

Today’s Puzzle:

Hugs are often abbreviated as “O” in letters and notes, and kisses are abbreviated as “X”. This puzzle can be a Valentine’s card and will send hugs and kisses to whomever you give it.

Can you 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 1587:

1587 is divisible by 3 because 1 + 5 + 8 + 7 = 21, a number divisible by 3.
1587 ÷ 3 = 529. Perhaps you have memorized some perfect squares and remember that 23² = 529. (If your parents invest in a 529 college plan for your education, you are more likely to have a college degree by the time you are 23. That’s how I remember it!)

  • 1587 is a composite number.
  • Prime factorization: 1587 = 3 × 23 × 23, which can be written 1587 = 3 × 23².
  • 1587 has at least one exponent greater than 1 in its prime factorization so √1587 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1587 = (√529)(√3) = 23√3.
  • The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 1587 has exactly 6 factors.
  • The factors of 1587 are outlined with their factor pair partners in the graphic below.

More about the Number 1587:

1587 is the difference of two squares three different ways:
794² – 793² = 1587,
266² – 263² = 1587, and
46² – 23² = 1587.

 

1577 Crossing a Bridge

Today’s Puzzle:

Merriam-Webster defines a bridge as “a structure carrying a pathway or roadway over a depression or obstacle (such as a river)”. Bridges have been built over wide rivers, but what type of bridge is needed when the obstacle is prejudice, hate, or greed?

Today is Martin Luther King Jr. Day in the United States. He tirelessly worked on such bridges. The movie, Selma, portrays much of what he and his companions did to cross a certain bridge in Alabama.  Although ALL of his protests were peaceful, some who followed him were brutally killed. Dr. King himself was assassinated when he was just 39 years old.

Today bridges still need to be built. Education can be an important bridge-builder. If you have Netflix, type “Martin Luther King” into the search, and several eye-opening movies and documentaries will appear. I highly recommend 13th, a documentary that shows how the same amendment to the constitution that abolished slavery has a loophole that essentially allows slavery to continue to this day. I was appalled.

The puzzle below is easy to solve. How to build bridges that make the world better for EVERYONE is a far more important and difficult puzzle that each of us needs to ponder to get closer to a solution.

Factors of 1577:

  • 1577 is a composite number.
  • Prime factorization: 1577 = 19 × 83.
  • 1577 has no exponents greater than 1 in its prime factorization, so √1577 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 1577 has exactly 4 factors.
  • The factors of 1577 are outlined with their factor pair partners in the graphic below.

More about the Number 1577:

1577 is the difference of two squares two different ways:
789² – 788² = 1577, and
51² – 32² = 1577.

1577 is the sum of the 2 numbers from 788 to 789.
1577 is the sum of the 19 numbers from 74 to 92. (83 is in the exact middle of that list of numbers.)
1577 is the sum of the 38 numbers from 23 to 60.