Prime factorization

1782 Don’t Chop Down This Factor Tree!

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Today’s Puzzle:

Today is Monday, February 19. In the United States, we are celebrating Presidents’ Day, honoring most especially two important presidents who were born in February.

Exactly one week ago was February 12.

George Washington was born on February 11, 1731, Julian calendar.
Abraham Lincoln was born on February 12, 1809, Gregorian calendar.

The Julian calendar didn’t have leap days, so in 1752 a year and eleven days were added to Washington’s birthday to convert it to the Gregorian calendar.

Neither president will ever have his birthday on the third Monday of February when Presidents’ Day is observed. Too bad the second Monday of February wasn’t chosen instead. Then we could fudge a little and say that Presidents’ Day would be observed on one of their birthdays 2/7 of the time!

What days of the month are the earliest and the latest that a second Monday could be? 

When I was young I was told the story about George Washington chopping down a cherry tree. When he was confronted, he would not and could not tell a lie, and confessed his misdeed. As I got older, I learned that this was a fabricated story designed to teach children honesty of all things!

Nevertheless, some people celebrate Presidents’ Day by eating a cherry pie in remembrance of that story.

Factors of 1782:

This is my 1782nd post. Since it’s Presidents’ Day, I thought I would make a few factor trees for that number. You could think of the prime factors in red as cherries on the trees. Notice that all the prime factors are low-hanging fruit on these particular trees!

  • 1782 is a composite number.
  • Prime factorization: 1782 = 2 × 3 × 3 × 3 × 3 × 11, which can be written 1782 = 2 × 3⁴ × 11.
  • 1782 has at least one exponent greater than 1 in its prime factorization so √1782 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1782 = (√22)(√81) = 9√22. The exponents in the prime factorization are 1, 4, and 1. Adding one to each exponent and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1782 has exactly 20 factors.
  • The factors of 1782 are outlined with their factor pair partners in the graphic below.

More About the Number 1782:

(5(27²) -3(27))/2 = 1782, so it is the 27th heptagonal number after 0.

Here’s another cool fact about 1782 from OEIS.org.

1781 A Mystery Puzzle for You to Solve

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Today’s Puzzle:

Is this mystery-level puzzle difficult or easy to solve? I’m not telling. You’ll have to try it for yourself to find out. As always, there is only one solution.

Factors of 1781:

1781 ÷ 4 leaves a remainder of 1, and 41² + 10² = 1781. Could 1781 be a prime number? It will be unless it has a prime number hypotenuse less than √1781 as a divisor. In other words, is it divisible by 5, 13, 17, 29, 37, or 41?

1781 obviously isn’t divisible by 5, and since it’s 41² + 10², it isn’t divisible by 41 either. That means we only have to check if it is divisible by 13, 17, 29, and 37.

So is it prime or composite?

  • 1781 is a composite number.
  • Prime factorization: 1781 = 13 × 137.
  • 1781 has no exponents greater than 1 in its prime factorization, so √1781 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 1781 has exactly 4 factors.
  • The factors of 1781 are outlined with their factor pair partners in the graphic below.

More About the Number 1781:

Not only does 41² + 10² = 1781, but
34² + 25² = 1781.

That 34² lets us know right away that 1781 is not divisible by 17, but any number that is the sum of two squares in more than one way is never a prime number.

1781 is the hypotenuse of FOUR Pythagorean triples:

531-1700-1781, calculated from 34² – 25², 2(34)(25), 34² + 25²,
685-1644-1781, which is (5-12-13) times 137,
820-1581-1781, calculated from 2(41)(10), 41² – 10², 41² + 10², and
1144-1365-1781, which is 13 times (88-105-137).

1781 is also the difference of two squares in two different ways:
891² – 890² = 1781, and
75² – 62² = 1781.

1780 Reflections of a Polygonal Bird

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Today’s Puzzle:

What ordered pairs were used to create this bird?

Its eye was formed from an equation of a circle:
(x – 7)²+ (y – 15)² = 3/4.

After creating the polygonal bird using ordered pairs and that circle equation, I wanted to do other things with the bird. Everything I did was like a puzzle for me to figure out.

Could I make it “fly”? Yes!

 

Could I make it reflect itself more than once over the y-axis and the x-axis? Yes! And I could make it do some sliding at the same time!

This next one was the toughest for me to do. I wanted the bird to be in motion rotating counter-clockwise around the origin. I was able to do it, but Desmos wouldn’t save the sliders exactly the way I wanted. I will need your help on this one. Click on this rotating bird link, then push play on slider a. About the time that slider goes to zero, push play on slider b. If you hit the sliders just right, it will look something like this GIF I made, but slower:

Rotating Polygonal Birds

make science GIFs like this at MakeaGif

 

Factors of 1780:

Perhaps our polygonal bird would like to fly to a tree. Here’s a factor tree for 1780 that it can take a rest on.

I knew that 1780 was divisible by 4 because its last two digits are divisible by 4.

  • 1780 is a composite number.
  • Prime factorization: 1780 = 2 × 2 × 5 × 89, which can be written 1780 = 2² × 5 × 89.
  • 1780 has at least one exponent greater than 1 in its prime factorization so √1780 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1780 = (√4)(√445) = 2√445.
  • 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 1780 has exactly 12 factors.
  • The factors of 1780 are outlined with their factor pair partners in the graphic below.

More About the Number 1780:

1780 is the difference of two squares in two different ways:
446² – 444² = 1780, and
94² – 84² = 1780.

1780 is the sum of two squares in two different ways:
42² + 4² = 1780, and
36² + 22² = 1780.

1780 is the hypotenuse of four Pythagorean triples:
336-1748-1780, calculated from 2(42)(4), 42² – 4², 42² + 4²,
780-1600-1780, which is 20 times (39-80-89)
812-1584-1780, calculated from 36² – 22², 2(36)(22), 36² + 22², and
1068-1424-1780, which is (3-4-5) times 356.

1780 is KK in base 88 because
20(88) + 20(1) = 20(89) = 1780.

1779 How Many Similar Triangles Are There in This Image?

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Today’s Puzzle:

All of the triangles in the image below are similar. How many similar triangles are there in the image? Why are they similar? Hint: If I were counting them, I would list all the triangles by writing each one indicating the sides in this order every time: the smallest, the medium, and the longest side. Don’t forget to list ΔLKJ. It’s pretty tiny!

Factors of 1779:

  • 1779 is a composite number.
  • Prime factorization: 1779 = 3 × 593.
  • 1779 has no exponents greater than 1 in its prime factorization, so √1779 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 1779 has exactly 4 factors.
  • The factors of 1779 are outlined with their factor pair partners in the graphic below.

More About the Number 1779:

1779 is the hypotenuse of a Pythagorean triple:
1104-1395-1779, which is 3 times (368-465-593).

From OEIS.org we learn that 1779 = 10,016,218,555,281, and that’s the smallest 4th power that has 14 digits.

1779 is palindrome 323 in base 24 because
3(24²) + 2(24) + 3(1) = 1779.

1778 Happy Valentine’s Day!

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Today’s Puzzle:

I ♥ U. Here’s a Valentine’s Day puzzle for you to enjoy. It might be a little tricky so remember to use logic to find all the factors! There are some other mathy Valentine’s Day activities at the end of the post.

Factors of 1778:

 

  • 1778 is a composite number.
  • Prime factorization: 1778 = 2 × 7 × 127.
  • 1778 has no exponents greater than 1 in its prime factorization, so √1778 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 1778 has exactly 8 factors.
  • The factors of 1778 are outlined with their factor pair partners in the graphic below.

More About the Number 1778:

1778 is the sum of four consecutive numbers:
443 + 444 + 445 + 446 = 1778.

1778 is the sum of seven consecutive numbers:
251 + 252 + 253 + 254 + 255 + 256 + 257 = 1778.

1778 is not the difference of two squares, but it is this:
446² – 445² + 444² – 443² =  1778.

1778 is palindrome, A6A in base13, because
10(13²) + 6(13) + 10(1) = 1778.

Other Mathy Valentine’s Day Activities:

 

1775 and Cupid’s Arrow

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Today’s Puzzle:

Will Cupid’s Arrow hit you right in your heart this year? Who knows? Solving this puzzle might help! It’s a level 3 puzzle so begin with the clues in the top row, then work your way down the puzzle row by row until you have found all the factors.

Factors of 1775:

  • 1775 is a composite number.
  • Prime factorization: 1775 = 5 × 5 × 71, which can be written 1775 = 5² × 71.
  • 1775 has at least one exponent greater than 1 in its prime factorization so √1775 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1775 = (√25)(√71) = 5√71.
  • 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 1775 has exactly 6 factors.
  • The factors of 1775 are outlined with their factor pair partners in the graphic below.

More About the Number 1775:

1775 is the difference of two squares in three different ways:
888² – 887² = 1775,
180² – 175² = 1775, and
48² – 23² = 1775.

1775 is the hypotenuse of two Pythagorean triples:
497-1704-1775, calculated from (7-24-25) times 71, and
1065 1420 1775, calculated from (3-4-5) times 355.

From OEIS.org, we learn that 1775 is one of the numbers in this Fibonacci-like series:
1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, . . .
Did you notice that 1+7=8, 7+8=15, and so forth? That’s why it’s called a Fibonacci-like series.

1775 is the repdigit PP in base 70. P is the 25th number in base 70. Thus,
25(70) + 25(1) = 25(71) = 1775.

1773 You Will L♥ve This Multiplication Table Puzzle!

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Today’s Puzzle:

It’s almost Valentine’s Day! Enjoy this heart-shaped multiplication table puzzle! You only need to know one set of ten math facts to complete this puzzle, but which set is it? The two’s? the three’s? the four’s? or something different? You CAN figure it out, so give it a try! There is only one solution.

Factors of 1773:

Does 1+7+7+3 = a number divisible by 3? I’ve played enough cribbage to know instantly that 1+7+7=15. Add the remaining 3 to the 15, and we get 18, a number divisible by both 3 and 9, so 1773 is divisible by both 3 and 9.

  • 1773 is a composite number.
  • Prime factorization: 1773 = 3 × 3 × 197, which can be written 1773 = 3² × 197.
  • 1773 has at least one exponent greater than 1 in its prime factorization so √1773 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1773 = (√9)(√197) = 3√197.
  • 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 1773 has exactly 6 factors.
  • The factors of 1773 are outlined with their factor pair partners in the graphic below.

More About the Number 1773:

1773 is the sum of two squares:
42² + 3² =1773.

1773 is the hypotenuse of one Pythagorean triple:
252-1755-1773, calculated from 2(42)(3), 42² – 3², 42² + 3².
It is also 9(28-195-197).

1773 is palindrome 909 in base 14 because
9(14²) + 0(14) + 9(1) = 1773.

1770 This Christmas, Don’t Let the Taxman Get Most of Your Cash!

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Today’s Puzzle:

1770 = 30 · 59.
1770 = (60 · 59)/2.

That means 1770 is a triangular number. If we have 59 envelopes numbered 1 to 59, and each envelope contained the amount of money on the outside of the envelope, we would have $1770 in cash at stake. In this game, the TAXMAN wants to take as much money as he can get, but you control how much he can take: Can you allow him to get as little as possible? 

You can play Taxman easily with these printable Taxman “envelopes” and Taxman Scoring Calculator because each “envelope” lists all the factors of the envelope number. Your first selection should be the biggest prime number on the board because then the only envelope the taxman can get on that turn is the 1 envelope. The Taxman must be able to take at least one envelope on every turn. Try to make it so he can only get one or at most two envelopes on each turn. When it is no longer possible for you to take an envelope that allows the Taxman to take at least one envelope, too, the taxman gets ALL the rest of the envelopes. You win if you can keep more than half of your cash. Good luck!

Factors of 1770:

  • 1770 is a composite number.
  • Prime factorization: 1770 = 2 × 3 × 5 × 59.
  • 1770 has no exponents greater than 1 in its prime factorization, so √1770 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 1770 has exactly 16 factors.
  • The factors of 1770 are outlined with their factor pair partners in the graphic below.

More About the Number 1770:

As mentioned earlier 1770 is the 59th triangular number because 59(60)/2 = 1770.

1770 is also the 30th hexagonal number because 30(2·30-1) = 1770. (Every hexagonal number is also a triangular number.)

I’ve made images of hexagonal numbers before, but this time I wanted to make one using this hexagon template:

1770 is the hypotenuse of a Pythagorean triple:
1062-1416-1770, which is (3-4-5) times 354.

1770 is repdigit, UU, in base 58 because
30(58) + 30(1) = 30(58 + 1) = 30(59) = 1770.

1765 On This Memorial Day

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Today’s Puzzle:

This weekend I laid a bouquet of red and white flowers on my husband’s grave and decided to make a red rose Memorial Day puzzle for the blog as well. It is a mystery-level puzzle.

Write the number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. There is only one solution.

Factors of 1765:

  • 1765 is a composite number.
  • Prime factorization: 1765 = 5 × 353.
  • 1765 has no exponents greater than 1 in its prime factorization, so √1765 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 1765 has exactly 4 factors.
  • The factors of 1765 are outlined with their factor pair partners in the graphic below.

More About the Number 1765:

1765 is the sum of two squares in two different ways:
42² + 1² = 1765, and
33² + 26² = 1765.

1765 is the hypotenuse of FOUR Pythagorean triples:
84 1763 1765, calculated from 2(42)(1), 42² – 1², 42² + 1²,
413 1716 1765, calculated from 33² – 26², 2(33)(26), 33² + 26²,
1059-1412-1765, which is (3-4-5) times 353, and
1125-1360-1765, which is 5 times (225-272-353).

1765 is a digitally powerful number:
1⁴ + 7³ + 6⁴ + 5³ = 1765.

1765 is a palindrome in a couple of different bases:
It’s A5A base 13 because 10(13²) + 5(13) + 10(1) = 1765, and
it’s 1D1 base 36 because 1(36²) + 13(36) + 1(1) = 1765.

1763 Daffodil Puzzle

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Today’s Puzzle:

Spring has sprung and perhaps flowers are blooming in your area. I think my favorite flowers are daffodils. I love the way they are shaped and their vibrant colors.

This daffodil puzzle is a great way to welcome spring. It may be a little bit tricky, but I think if you carefully use logic you will succeed! Just write each of the numbers 1 to 12 in the first column and again in the top row so that those numbers are the factors of the given clues. As always there is only one solution.

Here’s the same puzzle if you’d like to print it using less ink:

Factors of 1763:

  • 1763 is a composite number and the product of twin primes.
  • Prime factorization: 1763 = 41 × 43.
  • 1763 has no exponents greater than 1 in its prime factorization, so √1763 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 1763 has exactly 4 factors.
  • The factors of 1763 are outlined with their factor pair partners in the graphic below.

More About the Number 1763:

1763 is the difference of two squares in two different ways:
882² – 881² = 1763, and
42² – 1² = 1763. (That means the next number will be a perfect square!)

1763 is the hypotenuse of a Pythagorean triple:
387-1720-1763, which is (9-40-41) times 43.

1763 is palindrome 3E3 in base 22 because
3(22²) + 14(22) + 3(1) = 1763.

Lastly and most significantly: 15, 35, 143, 323, 899, and 1763 begin the list of numbers that are the product of twin primes. 1763 is just the sixth number on that list! If we include the products of two consecutive primes whether they are twin primes or not, the list is still fairly small. How rarely does that happen?

When it was 2021, did you realize how significant that year was?