1708 Happy Birthday, Jo Morgan!

Today’s Puzzle:

A few days ago I published a new kind of factoring puzzle. Jo Morgan of Resourceaholic.com, keeps an eye out for new mathematical resources on Twitter. She was one of the first to notice and like my puzzle. Because of her, lots of other people noticed the puzzle, too. Today is Jo’s birthday, and I decided to make a similar puzzle for her to enjoy. You might find it slightly more difficult than the earlier puzzle, but use logic from the beginning, and you will be able to fit in all the factors.

Factors of 1708:

  • 1708 is a composite number.
  • Prime factorization: 1708 = 2 × 2 × 7 × 61, which can be written 1708 = 2² × 7 × 61.
  • 1708 has at least one exponent greater than 1 in its prime factorization so √1708 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1708 = (√4)(√427) = 2√427.
  • 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 1708 has exactly 12 factors.
  • The factors of 1708 are outlined with their factor pair partners in the graphic below.

More About the Number 1708:

1708 is the hypotenuse of a Pythagorean triple:
308-1680-1708, which is 28 times (11-60-61).

1707 Subtraction Distraction

Today’s Puzzle:

Even though this Subtraction Distraction puzzle has more boxes than the one I published a couple of weeks ago, it is actually an easier puzzle. Can you write the numbers 1 to 12 in the boxes so that each triangle is its adjacent left box minus its adjacent right box?

Factors of 1707:

1 + 7 + 0 + 7 = 15, a multiple of 3, so 1707 is divisible by 3.

  • 1707 is a composite number.
  • Prime factorization: 1707 = 3 × 569.
  • 1707 has no exponents greater than 1 in its prime factorization, so √1707 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 1707 has exactly 4 factors.
  • The factors of 1707 are outlined with their factor pair partners in the graphic below.

More About the Number 1707:

1707 is the hypotenuse of a Pythagorean triple:
693-1560-1707, which is 3 times (231-520-569).

1706 Can You Make the Factors Fit?

Today’s Puzzle:

It’s 2022. Happy New Year! I wanted to make a puzzle that has 20 and 22 as clues and thought about what I could do.

I continue to be inspired by an old addition puzzle Sarah Carter @mathequalslove shared on Twitter:

I decided to tweak that puzzle into a multiplication puzzle. I ran into a problem, however. Having products in every triangle made the puzzle way too easy. How do I fix that? I removed some of the product clues. Can you use logic and factoring to know where each factor from 1 to 12 belongs? Can you determine the missing products? I hope you have lots of fun finding the puzzle’s only solution! And I hope you make the factors fit instead of having a fit trying!

Here’s something I haven’t told you before: I made lots of multiplication-table puzzles years before I started blogging. I wanted to give the puzzles a good name. At first, I called them “Turn the Tables on Multiplication” or “Turn the Tables” for short. I thought that title was clever but a little bit unwieldy. For a short time, I called the puzzles “Factor Fits.” It was a play on words because all the factors fit, but they might give you fits as you try to find them. I finally settled on “Find the Factors.” That title doubled as instructions for the puzzles. I still liked the name “Factor Fits,” and this puzzle lets me give new life to that name.

Factors of 1706:

  • 1706 is a composite number.
  • Prime factorization: 1706 = 2 × 853.
  • 1706 has no exponents greater than 1 in its prime factorization, so √1706 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 1706 has exactly 4 factors.
  • The factors of 1706 are outlined with their factor pair partners in the graphic below.

More About the Number 1706:

1706 is the sum of two squares:
41² + 5² = 1706.

1706 is the hypotenuse of a Pythagorean triple:
410-1656-1706, calculated from 2(41)(5), 41² – 5², 41² + 5².
It is also 2 times (205-828-853).

2022 Facts and Factors

Today’s Puzzle:

Solve these two multiplication problems and see how mirror-like they are!

Only 50 numbers less than 10000 can make a similar claim to fame:

Countdown to 2022:

Early in 2021, I found a countdown equation for 2022 that also involves its factors. A couple of weeks ago, I found another one. I can’t decide which one I like the best. Even though you can only focus on one equation at a time, the countdown will show both of them concurrently:

2022 Countdown Equations

make science GIFs like this at MakeaGif
Here is a still of the last frame:

 

Factors of 2022:

You might be asked to find the factors of 2022 several times in the coming year.
It’s an even number, so it’s divisible by 2.
It’s also easy to remember that it is divisible by 3 because 2 + 2 + 2 = 6, a multiple of 3.
This graphic may help you remember that it is divisible by 337:
  • 2022 is a composite number.
  • Prime factorization: 2022 = 2 × 3 × 337.
  • 2022 has no exponents greater than 1 in its prime factorization, so √2022 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 2022 has exactly 8 factors.
  • The factors of 2022 are outlined with their factor pair partners in the graphic below.

More About the Number 2022:

2022 is in the hypotenuse of a Pythagorean triple:
1050-1728-2022, which is 6 times (175-288-337).

It is also the short leg in three other Pythagorean triples:
2022-2696-3370, which is (3-4-5) times 674,
2022-340704-340710, which is 6 times (337-56784-56785), and
2022-1022120-1022122, which is 2 times (1011-511060-511061)

2022 uses only 2’s and 0’s in base 10 and in base 3:
2022₁₀ = 2202220₃.

2022 is the sum of positive consecutive numbers in three different ways:

2022 Magic Squares:

All of the above are the facts that I came up with. Lots of people on Twitter have found other facts about 2022. Be sure to check them out!

Tweets Celebrating 2022’s Mathematical Properties:

Here are some tweets about 2022 that I’ve seen on Twitter. I’ll add more as I see them. They are not listed in order of difficulty, but more or less, in the order that I saw them.

 


I’ll end with this thread that you’ll just have to click on to appreciate. It’s packed with tons of mathematical reasons 2022 will a fabulous year!

1704 Christmas Factor Tree

Today’s Puzzle:

If you know the factors of the clues in this Christmas tree, and you use logic, it is possible to write each number from 1 to 12 in both the first column and the top row to make a multiplication table. It’s a level six puzzle, so it won’t be easy, even for adults, but can YOU do it?

Factors of 1704:

If you were expecting to see a factor tree for the number 1704, here is one of several possibilities:

  • 1704 is a composite number.
  • Prime factorization: 1704 = 2 × 2 × 2 × 3 × 71, which can be written 1704 = 2³ × 3 × 71.
  • 1704 has at least one exponent greater than 1 in its prime factorization so √1704 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1704 = (√4)(√426) = 2√426.
  • 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 1704 has exactly 16 factors.
  • The factors of 1704 are outlined with their factor pair partners in the graphic below.


More About the Number 1704:

1704 is the difference of two squares in FOUR different ways:
427² – 425² = 1704,
215² – 211² = 1704,
145² – 139² = 1704, and
77² – 65² = 1704.

Why was Six afraid of Seven? Because Seven ate Nine.
1704 is 789 in a different base:
1704₁₀ = 789₁₅ because 7(15²) + 8(15¹) + 9(15º) = 1704.

1702 A Puzzle Idea from @mathequalslove Tweaked into a Subtraction Puzzle That Directs You to a Post from NebusResearch

Today’s Puzzle:

Joseph Nebus is nearly finished with all the posts in his Little 2021 Mathematics A to Z series. Every year he requests that his readers give him mathematical subjects to write about. At my suggestion, he recently wrote about subtraction, and how it is a subject that isn’t always as elementary as you might expect.  With a touch of humor, we learn that subtraction opens up whole new topics in mathematics.

I wanted to make a puzzle to commemorate his post. I gave it some thought and remembered a tweet from Sarah Carter @mathequalslove:

That puzzle originated from The Little Giant Encylopedia of Puzzles by the Diagram Group. I wondered how the puzzle would work if it were a subtraction puzzle instead of an addition puzzle, and here’s how I tweaked it:

 

There is only one solution. I hope you will try to find it! If you would like a hint, I’ll share one at the end of this post.

Factors of 1702:

  • 1702 is a composite number.
  • Prime factorization: 1702 = 2 × 23 × 37.
  • 1702 has no exponents greater than 1 in its prime factorization, so √1702 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 1702 has exactly 8 factors.
  • The factors of 1702 are outlined with their factor pair partners in the graphic below.

More About the Number 1702:

1702 is the hypotenuse of a Pythagorean triple:
552-1610-1702, which is (12-35-37) times 46.

1702² = 2896804, and
2197² = 4826809.
Do you notice what OEIS.org noticed about those two square numbers?

Puzzle Hint:

Here’s how I solved the puzzle: I let the rightmost box be x. Then using the values in the adjacent triangles and working from right to left, I wrote the values of the other boxes in terms of x.

x – 5 went in the box that is second to the right,
x – 5 + 2 = x – 3 went in the next box,
x – 3 + 5 = x + 2,
x + 2 – 6 = x – 4,
x – 4 + 5 = x + 1, and so on until I had assigned a value in terms of x for every box.

Think about it, and this hint should be enough for you to figure out where the numbers from 1 to 9 need to go.

1701 Is a Decagonal Number

Today’s Puzzle:

There is a pattern to the decagonal numbers. Can you figure out what it is?

Factors of 1701:

1701 is divisible by nine because 1 + 7 + 0 + 1 = 9.

  • 1701 is a composite number.
  • Prime factorization: 1701 = 3 × 3 × 3 × 3 × 3 × 7, which can be written 1701 = 3⁵ × 7.
  • 1701 has at least one exponent greater than 1 in its prime factorization so √1701 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1701 = (√81)(√21) = 9√21.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1701 has exactly 12 factors.
  • The factors of 1701 are outlined with their factor pair partners in the graphic below.

More About the Number 1701:

1701 is the difference of two squares in SIX different ways.
851² – 850² = 1701,
285² – 282² = 1701,
125² – 118² = 1701,
99² – 90² = 1701,
51² – 30² = 1701, and
45² – 18² = 1701.

1701 is the 21st decagonal number because
21(4·21 – 3) =
21(84-3) =
21(81) = 1701.

There is decagonal number generating function:
x(7x+1)/(1-x)³ = x + 10x² + 27x³ + 52x⁴ + 85x⁵ + . . .

The 21st term of that function is 1701 x²¹.

 

1699 Sweet Candy Cane

Today’s Puzzle:

Solving this candy cane puzzle can be a sweet experience. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Factors of 1699:

  • 1699 is a prime number.
  • Prime factorization: 1699 is prime.
  • 1699 has no exponents greater than 1 in its prime factorization, so √1699 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1699 has exactly 2 factors.
  • The factors of 1699 are outlined with their factor pair partners in the graphic below.

How do we know that 1699 is a prime number? If 1699 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1699. Since 1699 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1699 is a prime number.

More About the Number 1699:

1699 is the third prime in a prime triple. What are the other two primes in that triple?

1699 is the difference of two squares:
850² – 849² = 1699.

1698 A Little Virgács and Candy

Today’s Puzzle:

If you were a child in Hungary, you might have found some virgács and some candy in your boot this morning. Mikulás (St. Nick) would have given you the candy because of how good you’ve been this year, and the virgács for those times you weren’t so good.

This virgács and candy puzzle is like a mixed-up multiplication table. It is a lot easier to solve because I made it a level 3 puzzle. First, find the common factor of 56 and 72 that will allow only numbers between 1 and 12 to go in the first column. Put the factors in the appropriate cells, then work your way down the puzzle, row by row until each number from 1 to 12 is in both the first column and the top row.

Factors of 1698:

  • 1698 is a composite number.
  • Prime factorization: 1698 = 2 × 3 × 283.
  • 1698 has no exponents greater than 1 in its prime factorization, so √1698 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 1698 has exactly 8 factors.
  • The factors of 1698 are outlined with their factor pair partners in the graphic below.

More About the Number 1698:

1698 = 2(849)(1), so it is a leg in the Pythagorean triple calculated from
2(849)(1), 849² – 1², 849² + 1².

1698 = 2(283)(3), so it is a leg in the Pythagorean triple calculated from
2(283)(3), 283² – 3², 283² + 3².

1697 A Boot in the Window

Today’s Puzzle:

Tonight throughout many parts of the world children will place their polished boots in a window awaiting a visit from St. Nick. In the morning they will find their boots filled with favorite candies.

You can solve this boot puzzle by writing the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will become the start of a multiplication table.

Factors of 1697:

  • 1697 is a prime number.
  • Prime factorization: 1697 is prime.
  • 1697 has no exponents greater than 1 in its prime factorization, so √1697 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1697 has exactly 2 factors.
  • The factors of 1697 are outlined with their factor pair partners in the graphic below.

How do we know that 1697 is a prime number? If 1697 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1697. Since 1697 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1697 is a prime number.

More About the Number 1697:

1697 is the sum of two squares:
41² + 4² = 1697.

1697 is the hypotenuse of a Pythagorean triple:
328-1665-1697, calculated from 2(41)(4), 41² – 4², 41² + 4².

Here’s another way we know that 1697 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 41² + 4² = 1697 with 41 and 4 having no common prime factors, 1697 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1697. Since 1697 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1697 is a prime number.