Factors

2022 Facts and Factors

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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

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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

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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

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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

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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.

 

1700 Time for a Horse Race!

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Today’s Puzzle:

I’ve made a table of all the numbers from 1601 to 1700, their prime factorizations, and how many factors each of those numbers has.

Each number from 1601 to 1700 has 2, 3, 4, 6, 8, 10, 12, 16, 18, 24, 30 or 40 factors. Do more numbers have 2 factors, 3 factors, or a different number? Sure, you could use the table to count, but it will be more fun if we make a horse race out of it. What pony will you pick, 2, 3, 4, 6? Make your prediction of the winner, and then watch the horse race to see if you were right.

Here are all the horses lined up at the gate. Pick your pony!

And they’re off!

1601 to 1700 Horse Race

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The first half of the race was VERY interesting to me. Three different horses held the lead, and the leaders in the race was even neck and neck some of the time. After you know the winner, watch the race again, following the winner from start to finish.

Factors of the Number 1700:

  • 1700 is a composite number.
  • Prime factorization: 1700 = 2 × 2 × 5 × 5 × 17, which can be written 1700 = 2² × 5² × 17.
  • 1700 has at least one exponent greater than 1 in its prime factorization so √1700 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1700 = (√100)(√17) = 10√17.
  • 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 1700 has exactly 18 factors.
  • The factors of 1700 are outlined with their factor pair partners in the graphic below.

More About the Number 1700:

1700 is the sum of two squares THREE different ways:
40² + 10² = 1700,
38² + 16² = 1700, and
32² + 26² = 1700.

Those sum of two squares mean 1700 is the hypotenuse of some Pythagorean triples, SEVEN to be exact:

  1. 260-1680-1700, which is 20 times (13-84-85),
  2. 348-1664-1700, which is 4 times (87-416-425), but it can also be calculated from 32² – 26², 2(32)(26), 32² + 26²
  3. 476-1632-1700, which is (7-24-25) times 68,
  4. 720-1540-1700, which is 20 times (36-77-85),
  5. 800-1500-1700, which is (8-15-17) times 100, but it can also be calculated from 2(40)(10), 40² – 10², 40² + 10²,
  6. 1020-1360-1700, which is (3-4-5) times 340,
  7. 1188-1216-1700, which is 4 times (297-304-425), but it can also be calculated from 38² – 16², 2(38)(16), 38² + 16².

As OEIS.org informs us, 1700 is a Catalan number. It is found in the C(13,4) position, as shown below. With the exception of the lone 1 in the 0th row of the triangle, every number in Catalan’s triangle is the sum of the number above it and the number to its left. For example, 1700 = 1260 + 440.

I hope you have enjoyed learning about the number 1700.

1699 Sweet Candy Cane

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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

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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

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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.

1696 Inverses

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Today’s Puzzle:

Joseph Nebus of Nebusresearch recently wrote an extensive post about inverses at my request. Although inverses are often fascinating, advanced topics in mathematics, they can also be quite simple. For example, solving this puzzle will involve using the inverse of multiplication, division, AND it is the simplest division possible this time. Since it is December, I made this puzzle look like a gift for you. Just write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues could become a multiplication table.

For any gift you receive, you can invoke inverse exploring questions such as “how was this gift made?”

If you want to print the puzzle without color, here it is:

Factors of 1696:

  • 1696 is a composite number.
  • Prime factorization: 1696 = 2 × 2 × 2 × 2 × 2 × 53, which can be written 1696 = 2⁵ × 53.
  • 1696 has at least one exponent greater than 1 in its prime factorization so √1696 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1696 = (√16)(√106) = 4√106.
  • 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 1696 has exactly 12 factors.
  • The factors of 1696 are outlined with their factor pair partners in the graphic below.

More About the Number 1696:

1696 is the sum of two squares:
36² + 20² = 1696.

1696 is also the hypotenuse of a Pythagorean triple:
896-1440 -1696, which is 32 times (28-45-52).
It can also be calculated from 36² – 20², 2(36)(20), 36² + 20².

1696 is also the difference of two squares in FOUR different ways:
425² – 423² = 1696,
214² – 210² = 1696,
110² – 102² = 1696, and
61² – 45² = 1696.
I found those equations by using the even factor pairs of 1696 and taking the inverse of the fact that a² – b² = (a + b)(a – b).

1695 If You Can Solve 3×3 and 5×5 Magic Squares, Then You Can Solve a 15×15 Magic Square!

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Today’s Puzzle:

Completing a 15 × 15 magic square may seem daunting, but I assure you, if you can solve a 3 × 3 magic square and a 5 × 5 magic square, then you can complete a 15 × 15 magic square!

The 15 × 15 magic square below is made with twenty-five 3 × 3 magic squares. See the most famous 3 × 3 square in yellow? Do you see that the first 25 multiples of 9 are along the diagonals I’ve drawn? Do you understand the pattern that was used to make this magic square? Study it, then without looking at this one, can you make your own? Open this Excel file, 12 Factors 1683-1695, enable editing, and the sums of each row, column, and diagonal will automatically populate as you type in the numbers.

This next 15 × 15 magic square is made with nine 5 × 5 magic squares. The one in yellow is one of MANY possible 5 × 5 squares. Do you see the first nine multiples of 25 along the diagonals I’ve drawn? Once you understand this pattern, perhaps you would like to take a turn duplicating it. Try it yourself on that same Excel file, 12 Factors 1683-1695.

Factors of 1695:

  • 1695 is a composite number.
  • Prime factorization: 1695 = 3 × 5 × 113.
  • 1695 has no exponents greater than 1 in its prime factorization, so √1695 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 1695 has exactly 8 factors.
  • The factors of 1695 are outlined with their factor pair partners in the graphic below.

More About the Number 1695:

Since 1695 is the sum of three consecutive numbers, it is the magic sum of a particular 3 × 3 Magic Square. Those three consecutive numbers are in yellow in the square below.

Also, since 1695 is the sum of the five consecutive numbers shown in yellow below, it is the magic sum of this 5 × 5 magic square:

And lastly, 1695 is the sum of the 15 consecutive numbers shown in yellow below, so here is yet another 15 × 15 magic square with 1695 as the magic sum. This magic square uses the same pattern that works for all odd number magic squares. It is so satisfying to complete it yourself. Study this one and then give it a try! Open that same Excel file, 12 Factors 1683-1695, to make Excel to all the adding for you.

1695 is also the hypotenuse in FOUR Pythagorean triples:
225-1680-1695, which is 15 times (15-112-113),
828-1479-1695, which is 3 times (276-493-565)
1017-1356-1695, which is (3-4-5) times 339, and
1188-1209-1695, which is 3 times (396-403-565).

Did you notice that 565, 339, or 113 was a center number in every magic square in this post?

I hope you enjoy completing these magic squares on your own as you explore the number 1695.