1549 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write.

Factors of 1549:

  • 1549 is a prime number.
  • Prime factorization: 1549 is prime.
  • 1549 has no exponents greater than 1 in its prime factorization, so √1549 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1549 has exactly 2 factors.
  • The factors of 1549 are outlined with their factor pair partners in the graphic below.

How do we know that 1549 is a prime number? If 1549 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1549. Since 1549 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1549 is a prime number.

More about the number 1549:

1549 is the sum of two squares:
35² + 18² = 1549

1549 is the hypotenuse of a Pythagorean triple:
901-1260-1549, calculated from 35² – 18², 2(35)(18), 35² + 18².

Here’s another way we know that 1549 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 18² = 1549 with 35 and 18 having no common prime factors, 1549 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1549. Since 1549 is not divisible by 5, 13, 17, 29, or 37, we know that 1549 is a prime number.

1548 Puzzling Gerrymandering Questions

Today’s Puzzle:

If you had two types of candy and four kids, this is a good way to divide the candy. But if you had two types of voters and four congressional districts, is it a fair way to determine those congressional districts?

Gerrymandering happens when congressional district boundaries are drawn to give an advantage to one political party over another. The green party might be given an advantage in the drawing above, but does simply dividing the graphic into four quadrants give an advantage to the yellow party?

One article complains that one of the worst examples of gerrymandering is a congressional district shaped like a duck, but it is unclear if that duck is keeping like-minded people together or keeping them apart.

or

It is unclear to us if like-minded people were grouped together, but we can be certain that it was very clear to those who drew the boundaries.

Each of the graphics above had 12 yellow sections and 36 green sections. If you think it is only fair to let like-minded people elect someone who thinks like them,  how should congressional boundaries be drawn if the 12 yellow sections and the 36 green sections look like this?

No matter how you draw the boundaries, green will be in the majority in each congressional district, and most likely the majority will choose each district’s representative. But if you believe that gerrymandering is justified to benefit like-minded people, should those people NOT be represented by a like-minded representative simply because they don’t live next to each other?

These are good questions to puzzle over. Denise Gaskins has created the Gerrymandering Project to help you manipulate a 10 by 10 map of your creation in a variety of ways. This project will help every voter and future voter understand the mathematics and the politics of drawing boundaries on a larger scale. Check it out!

Factors of 1548:

  • 1548 is a composite number.
  • Prime factorization: 1548 = 2 × 2 × 3 × 3 × 43, which can be written 1548 = 2² × 3² × 43
  • 1548 has at least one exponent greater than 1 in its prime factorization so √1548 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1548 = (√36)(√43) = 6√43
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1548 has exactly 18 factors.
  • The factors of 1548 are outlined with their factor pair partners in the graphic below.

Factor Tree for 1548:

The last two digits of 1548 are a multiple of 4, so 1548 is divisible by 4.
1 + 5 + 4 + 8 = 18, a multiple of 9, so 1548 is divisible by 9.

Here’s how I used those two facts to make an autumn factor tree for 1548:

More about the Number 1548:

1548 is the difference of two squares in three different ways:
388² – 386² = 1548,
132² – 126² = 1548, and
52² – 34²  = 1548.

 

1547 is a Hexagonal Pyramidal Number

Today’s Puzzle:

1547 is the 13th hexagonal pyramidal number. Looking at the graphic that I made below of 1547 tiny squares, can you determine what the prime factorization of 1547 is?

Factors of 1547:

  • 1547 is a composite number.
  • Prime factorization: 1547 = 7 × 13 × 17.
  • 1547 has no exponents greater than 1 in its prime factorization, so √1547 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 1547 has exactly 8 factors.
  • The factors of 1547 are outlined with their factor pair partners in the graphic below.

More about the Number 1547:

1547 is the hypotenuse of FOUR Pythagorean triples:
147-1540-1547, which is 7 times (21-220-221),
595-1428-1547, which is (5-12-13) times 119,
728-1365-1547, which is (8-15-17) times 91, and
980-1197-1547, which is 7 times (140-171-221).

1546 Celebrating My 7th Blogiversary

Today’s Puzzle:

Seven years ago today at 2:11 in the morning, I published my first blog post. It featured a Find the Factors 1 – 10 puzzle with only perfect square clues. In honor of that first post, I publish this puzzle, also with only perfect square clues. This one looks like an exclamation point because I have enjoyed creating puzzles and writing these posts for you so much these seven years!

How do you solve the puzzle? Use logic to write the numbers from 1 to 10 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1546:

  • 1546 is a composite number.
  • Prime factorization: 1546 = 2 × 773.
  • 1546 has no exponents greater than 1 in its prime factorization, so √1546 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 1546 has exactly 4 factors.
  • The factors of 1546 are outlined with their factor pair partners in the graphic below.

More about the Number 1546:

1546 is the sum of two squares:
39² + 5² = 1546.

1546 is the hypotenuse of a Pythagorean triple:
390-1496-1546, calculated from 2(39)(5), 39² – 5², 39² + 5².
It is also 2 times (195-748-773)

1545 The Complete Message Made with Puzzle Letters

The Puzzles:

The last six Find the Factors Puzzles spelled out this message: I voted. I like the quote from Larry J. Sabato, “Every election is determined by the people who show up.” My vote will matter, and so will yours.

You can also print these puzzles from this excel file: Taxman & 1537-1544

American politics has become rather ugly, but voting is still important.

I am thrilled that I was able to vote for one of these candidates for governor of Utah, and I am very happy that one of them will become my governor even if it isn’t the same person I voted for.

Factors of 1545:

  • 1545 is a composite number.
  • Prime factorization: 1545 = 3 × 5 × 309.
  • 1545 has no exponents greater than 1 in its prime factorization, so √1545 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 1545 has exactly 8 factors.
  • The factors of 1545 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1545:

1545 is the hypotenuse of a Pythagorean triple:
927-1236-1545, which is (3-4-5) times 309.

1544 Final Letter of the Message

Today’s Puzzle:

This is the sixth and final letter of the message that I made for you. It’s a mystery puzzle, but that doesn’t necessarily mean that’s it’s a difficult puzzle. Give it a try and think about why I sent the message.

Factors of 1544:

  • 1544 is a composite number.
  • Prime factorization: 1544 = 2 × 2 × 2 × 193, which can be written 1544 = 2³ × 193
  • 1544 has at least one exponent greater than 1 in its prime factorization so √1544 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1544 = (√4)(√386) = 2√386
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1544 has exactly 8 factors.
  • The factors of 1544 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1544:

1544 is the sum of two squares:
38² + 10² = 1544.

1544 is the hypotenuse of a Pythagorean triple:
760-1344-1544, calculated from 2(38)(10), 38² – 10², 38² + 10².
It is also 8 times (95-168-193).

1543 Another Letter

Today’s Puzzle:

My goal was to make this next letter of my message using as few clues in the puzzle as possible. You now have enough clues to solve this puzzle and know my message to you.

Factors of 1543:

  • 1543 is a prime number.
  • Prime factorization: 1543 is prime.
  • 1543 has no exponents greater than 1 in its prime factorization, so √1543 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1543 has exactly 2 factors.
  • The factors of 1543 are outlined with their factor pair partners in the graphic below.

How do we know that 1543 is a prime number? If 1543 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1543. Since 1543 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1543 is a prime number.

One More Fact about the Number 1543:

1543 is the sum of consecutive numbers as well as the difference of those same consecutive numbers but squared:
771 + 772 = 1543;
772² – 771² = 1543.
(1543 has that property because it’s an odd number greater than 1.)

1542 Fourth Letter of the Message

Today’s Puzzle:

This level one puzzle is the easiest of all the puzzles in my message to you. It might also be the letter that helps you know exactly what the message says.

Factors of 1542:

  • 1542 is a composite number.
  • Prime factorization: 1542 = 2 × 3 × 257.
  • 1542 has no exponents greater than 1 in its prime factorization, so √1542 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 1542 has exactly 8 factors.
  • The factors of 1542 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1542

1542 is the hypotenuse of a Pythagorean triple:
192-1530-1542, which is 6 times (32-255-257).

 

1541 is the 23rd Octagonal Number

Today’s Puzzle:

Into which geometric shape can you arrange 1541 tiny squares?

The answer is an octagon as illustrated below.

Can you divide that octagon into 6 triangles, 5 of them representing the 22nd triangular number and 1 of them representing the 23rd?

It is possible because 5(22·23)/2 + (23·24)/2 = 1541.

We can simplify the left side of the equal sign:
5(22·23)/2 + (23·24)/2 =
5(11·23) + (23·12) =
23(5·11) + 23(12) =
23(55 + 12) =
23(67) = 1541

Factors of 1541:

  • 1541 is a composite number.
  • Prime factorization: 1541 = 23 × 67.
  • 1541 has no exponents greater than 1 in its prime factorization, so √1541 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 1541 has exactly 4 factors.
  • The factors of 1541 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1541:

1541 is the difference of two squares in two different ways:
771² – 770² = 1541 (Not coincidently, 770 + 771 = 1541), and
45² – 22² = 1541.

How Much of My $1540 Will The Taxman Get?

The Taxman Game

You might think that the picture below is just the numbers from 1 to 55 arranged in a 5×11 rectangle.

It’s really PLAY MONEY, one bill in each denomination from 1 to 55. If you add up the total value of all this play money, you will get 1540 because the sum of the numbers from 1 to 55 is 1540.

Most of that 1540 in play money can be yours to keep if you outsmart THE TAXMAN.

You can take any bill that you want, but The Taxman has to get paid every time you do. The Taxman will get ALL the available factors of the bill you take. If there isn’t at least one bill on hand that is a factor of your chosen bill, then you can’t take that bill. When none of the bills left are factors of any of the other bills that are left, The Taxman gets ALL the remaining bills. The value of those bills could add up quickly, and that’s how The Taxman might outsmart you.

You can print this excel file to play the Taxman game: Taxman & 1537-1544 It has all the play money bills from 1 to 100, and all the factors are typed on the top of the bills!

The factors at the top of each bill make it possible to play the game even if you don’t know how to multiply or divide yet. My granddaughter learned to play the game sometime around her eighth birthday. Before tackling 55 bills, she learned what to do for a smaller number of bills. I told her:
If 1 is the only bill, The Taxman wins.
If 1, 2 are the only bills, you take the 2, The Taxman gets the 1, and you win.
If 1, 2, 3 are the only bills, the best you can do is a draw (1 + 2 = 3),
But if there are at least four bills, it is possible to win every time.

She played the game over and over adding one more bill to the game each time she played. She enjoyed it very much and played it over 20 times until bedtime required her to stop. Each time she played we talked about how she would arrange the cards in a rectangle. For example, if she used cards 1 – 6, she would look at the factors of 6 and decide on a 2 × 3 rectangle or a 1 × 6 rectangle.

Knowing I wanted to write this post about the 55th triangular number (1540), I had her jump to using 55 bills the next day, but I helped her with it a little. We didn’t want to just win the game, we wanted to make it so The Taxman would get as little money as possible. Here is the final order of what we took and what The Taxman got.

Laying it out like that made it easy to have do-overs when needed.

A Way to Keep Score When Playing Taxman:

When it was time to score the game, instead of adding up all the numbers to see if we took more than The Taxman, we found smaller sums that equaled as many of our choices as we could as illustrated below. For example,  47 + 6 = 53, so there was no need to add 47 + 6 to The Taxman’s total or to add 53 to ours.

I found it enjoyable finding all those sums, but it isn’t necessarily easy for a child to do. After arranging all those sums on the table, we were able to see that we scored
35 + 39 + 38 + 12 + 30 + 50 + 24 + 48 + 42 = 318 more than The Taxman did.

We can use algebra to figure out exactly how much of our 1540 in play money we get to keep.
If The Taxman got X of the play money, we kept X + 318 of it, and
X + X + 318 = 1540,
2X + 318 = 1540,
2X + 318 – 318 = 1540 – 318,
2X = 1222,
X = 611, that’s what The Taxman got.
We kept 611 + 318 = 929 of our play money.

What percentage of the play money did The Taxman get?
611/1540 ≈ 0.40 which is 40%. Wow, that’s a lot!

Factors of 1540:

  • 1540 is a composite number.
  • Prime factorization: 1540 = 2 × 2 × 5 × 7 × 11, which can be written 1540 = 2² × 5 × 7 × 11
  • 1540 has at least one exponent greater than 1 in its prime factorization so √1540 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1540 = (√4)(√385) = 2√385
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1540 has exactly 24 factors.
  • The factors of 1540 are outlined with their factor pair partners in the graphic below.

Factor Cake for 1540:

1540 has lots of prime factors with 11 being the greatest so it makes a lovely factor cake:

What Kind of Shape Is 1540 in?

I know that 1540 = 55×56÷2 so, 1540 is the 55th triangular number. Guess what? Since 55 is also a triangular number (55 = 10×11÷2), we could have arranged the play money bills in a triangle like this instead of a 5 × 11 rectangle.

1540 is not only the 55th triangular number, but it is also the sum of the first 20 triangular numbers:

1+3+6+10+15+21+28+36+45+55+66+78+91+105+120+136+153+171+190+210 = 1540.

If you stack those twenty triangular numbers from smallest to biggest, you will get something shaped like a tetrahedron. That’s why we say that 1540 is the 20th tetrahedral number. We can also use a formula for the 20th tetrahedral number:
20(21)(22)/6 = 1540.

1540 is also the 28th hexagonal number. (All hexagonal numbers are also triangular numbers. That’s why it’s the 2(28)-1 = 55th triangular number.) It is the 28th hexagonal number because 2(28²) – 28 = 1540.

1540 is the 20th decagonal number because 4(20²) – 3(20) = 1540.

I didn’t make a visual, but 1540 is also the19th centered nonagonal number.
That’s because
3 × 19 = 57 and (57 – 2)(57 – 1)/2 = 1540.

1540 is also the hypotenuse of a Pythagorean triple triangle:
924-1232-1540 which is (3-4-5) times 308.

1540 is quite the shape-shifter number, isn’t it!