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:

 

1777 A Different Heart

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Today’s Puzzle:

Every year I make some heart-shaped puzzles, but this heart is different: I haven’t used this design before. Can it win you over? Some of the clues are tricky, so make sure you use logic to find the one and only solution.

Factors of 1777:

  • 1777 is a prime number.
  • Prime factorization: 1777 is prime.
  • 1777 has no exponents greater than 1 in its prime factorization, so √1777 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1777 has exactly 2 factors.
  • The factors of 1777 are outlined with their factor pair partners in the graphic below.

How do we know that 1777 is a prime number? If 1777 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1777. Since 1777 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1777 is a prime number.

More About the Number 1777:

1777 is the sum of two squares:
39² + 16² = 1777.

1777 is the hypotenuse of a primitive Pythagorean triple:
1248-1265-1777, calculated from 2(39)(16), 39² – 16², 39² + 16²

Here’s another way we know that 1777 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 16² = 1777 with 39 and 16 having no common prime factors, 1777 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1777. Since 1777 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1777 is a prime number.

1777 looks interesting in some other bases:
It’s 12121 in base 6 because 1(6⁴) + 2(6³) + 1(6²) + 2(6¹) + 1(6º) = 1777,
2L2 in base 25, because 2(25²) + 21(25) + 2(1) = 1777, and
1B1in base 37, because 1(37²) + 11(37) + 1(1) = 1777.

1776 A Single Rosebud

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Today’s Puzzle:

The gift of a single red rose is a way to say, “I love you.” To me, a single red rosebud would be saying, “I love you, and my love for you is growing.” To all my faithful readers, I give you this single red rosebud:

Here is the same puzzle without any added color if you want to save on printer ink.

Factors of 1776:

Another way to show love is to plant a tree. How about we plant a factor tree? Since 1776 has twenty different factor pairs, MANY possible factor trees could be planted. I chose to base this one on the fun fact that 1776 = 4 · 444:

  • 1776 is a composite number.
  • Prime factorization: 1776 = 2 × 2 × 2 × 2 × 3 × 37, which can be written 1776 = 2⁴ × 3 × 37
  • 1776 has at least one exponent greater than 1 in its prime factorization so √1776 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1776 = (√16)(√111) = 4√111
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1776 has exactly 20 factors.
  • The factors of 1776 are outlined with their factor pair partners in the graphic below.

Did you notice all the repdigit factors of 1776 in the table?

More About the Number 1776:

1776 is the hypotenuse of a Pythagorean triple:
576-1680-1776, which is (12-35-37) times 48.

1776 looks interesting in some other bases:
It’s 5115 in base7 because 5(7³) + 1(7²) +1(7) + 5(1) = 1776, and uh oh!
OO in base73 because 24(73) + 24(1) = 24(74) = 1776.
We’ve run out of letters in the alphabet to use as numbers, but I
will note that 37(47) + 37(1) = 37(48) = 1776.

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.

1774 A Mostly Square Heart for You to Play With

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Today’s Puzzle:

The celebrated author of Math Play, Libo Valencia, recently wrote a post on how he uses mathplay to help his nine-year-old daughter learn the multiplication table. One of the playful things they did together was find objects around the house to represent several perfect squares. For example, they happened to have some small bright yellow hexagons in their house and they used six of them to show that six times six is thirty-six. If you don’t have any bright yellow hexagons at your place, you probably have some hexagon-shaped nuts and/or bolts you could use to show 6 × 6 = 36.

All but two of the clues in today’s puzzle are perfect squares, so I’m dedicating this puzzle to Libo’s daughter. Square number thirty-six is a clue three times in the puzzle. The rules of the puzzle won’t allow 6 × 6 to be the factors for all three of them, however. I’m sure you can figure the puzzle out, anyway. Just make sure you’re having fun doing it. There is only one solution.

Factors of 1774:

  • 1774 is a prime number.
  • Prime factorization: 1774 is prime.
  • 1774 has no exponents greater than 1 in its prime factorization, so √1774 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1774 has exactly 2 factors.
  • The factors of 1774 are outlined with their factor pair partners in the graphic below.

More About the Number 1774:

1774 is a palindrome in a couple of bases:
It’s 626 in base17 because 6(17²) + 2(17) + 6(1) = 1774, and
it’s 383 in base23 because 3(23²) + 8(23) + 3(1) =1774.

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.