factor pairs

2022 Facts and Factors

/ ivasallay / 1 Comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / 1 Comment

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

/ ivasallay / Leave a comment

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.

 

1698 A Little Virgács and Candy

/ ivasallay / Leave a comment

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.

1696 Inverses

/ ivasallay / Leave a comment

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!

/ ivasallay / Leave a comment

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.

1694 Football Game Day

/ ivasallay / Leave a comment

Today’s Puzzle:

Today all over the United States family and friends will gather to watch or play a game of football. If you would like to change things up a little, here’s a game ball for you to practice multiplication and division facts. Just write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues form a multiplication table. Some clues might be tricky, but enough of them aren’t that I am confident you can score with this football!

Here’s the same puzzle without any distracting color:

Factors of 1694:

  • 1694 is a composite number.
  • Prime factorization: 1694 = 2 × 7 × 11 × 11, which can be written 1694 = 2 × 7 × 11².
  • 1694 has at least one exponent greater than 1 in its prime factorization so √1694 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1694 = (√121)(√14) = 11√14.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1694 has exactly 12 factors.
  • The factors of 1694 are outlined with their factor pair partners in the graphic below.

More About the Number 1694:

From OEIS.org we learn that 1694³ = 4,861,163,384, a number that uses each of the digits 1, 3, 4, 5, and 8 exactly twice.

1692 A Pilgrim’s Belt to Unbuckle

/ ivasallay / Leave a comment

Today’s Puzzle:

The logic needed to unbuckle this Pilgrim’s belt puzzle has several interesting twists and turns in it. Even adults will find it a challenge. Guessing and checking will only frustrate you. Use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work like a multiplication table.

Here’s the same puzzle if you want to print it using less ink.

Factors of 1692:

  • 1692 is a composite number.
  • Prime factorization: 1692 = 2 × 2 × 3 × 3 × 47, which can be written 1692 = 2² × 3² × 47.
  • 1692 has at least one exponent greater than 1 in its prime factorization so √1692 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1692 = (√36)(√47) = 6√47.
  • 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 1692 has exactly 18 factors.
  • The factors of 1692 are outlined with their factor pair partners in the graphic below.
  • 1568 is a composite number.

More About the Number 1692:

1692 is the difference of two squares in three different ways:
424² – 422² = 1692,
144² – 138² = 1692, and
56² – 38² = 1692.

The square of 1692 looks a little interesting:
1692² = 2862864.

1690 Today Is My 8th Blogiversary!

/ ivasallay / 2 Comments

Today’s Puzzle:

I made this Crazy-8 puzzle to commemorate the 8th anniversary of my blog.

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work together like a multiplication table. Some of it might be a little tricky, so make sure you use logic on every step!

My eighth year of blogging has been amazing for me:

  1. Denise Gaskins has a Kickstarter going for her latest book, 312 Things to Do with a Math Journal. One of those 312 things will be journaling about some of my puzzles.
  2. I’ve also hosted her fabulous Math Education Blog Carnival and been featured when other bloggers hosted it.
  3. Bill Davidson interviewed me for his podcast, Centering the Pendulum. Although I’m not one of the many “Eureka Math Giants” he knows, my interview was included in the mix.

  4. In the spring, THREE different types of puzzles I’ve made were published in the Austin Chronicle.

  5. Also a BIG thank you to YOU, reading this right now. I really appreciate you and others who have taken the time to read my thoughts and solve my puzzles.

It’s been a wonderful year. NONE of those things would have happened if I didn’t write a blog. I feel quite fortunate and humbled by it all. I think I’ll go on for another eight years!

Factors of 1690:

  • 1690 is a composite number.
  • Prime factorization: 1690 = 2 × 5 × 13 × 13, which can be written 1690 = 2 × 5 × 13².
  • 1690 has at least one exponent greater than 1 in its prime factorization so √1690 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1690 = (√169)(√10) = 13√10.
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1690 has exactly 12 factors.
  • The factors of 1690 are outlined with their factor pair partners in the graphic below.

More About the Number 1690:

1690 is the sum of two squares in THREE different ways:
41² + 3² = 1690,
39² + 13² = 1690, and
31² + 27² = 1690.

1690 is the hypotenuse of SEVEN Pythagorean triples:
232 1674 1690, calculated from 31² – 27², 2(31)(27), 31² + 27²,
246 1672 1690, calculated from 2(41)(3), 41² – 3², 41² + 3²,
416 1638 1690, which is 26 times (16-63-65),
650 1560 1690, which is (5-12-13) times 130.
858 1456 1690, which is 26 times (33-56-65),
1014 1352 1690, calculated from 2(39)(13), 39² – 13², 39² + 13², but it is also (3-4-5) times 338, and
1190 1200 1690, which is 10 times (119-120-169).