Puzzle

Facts, Factors, and a Countdown to 2023

/ ivasallay / 3 Comments

Countdown to 2023:

The last 10 seconds of the year, we like to countdown from 10 to the new year. I like a mathematical way of counting down so I try to make an equation with the numbers from 10 to 1 that equals the coming year. This year I could have based my countdown  on last year’s countdown and said
(10-9+8×7×6)(5-4)(3)(2)+1 = 2023,
but this is a blog about factoring so I want a countdown that takes you to the prime factors of 2023 first. Here’s my countdown: (Note: Even though I used 1 as a factor twice in the countdown, I am very much aware that 1 is not a prime factor of any number.)

Countdown to 2023

make science GIFs like this at MakeaGif
Here’s the final shot from the Countdown:

A Puzzle for 2023:

2023 has something in common with 2022. When either number and their reverses are squared, something interesting happens…it’s almost like looking in a mirror!

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

Factors of 2023:

  • 2023 is a composite number.
  • Prime factorization: 2023 = 7 × 17 × 17, which can be written 2023 = 7 × 17².
  • 2023 has at least one exponent greater than 1 in its prime factorization so √2023 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √2023 = (√289)(√7) = 17√7.
  • The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 2023 has exactly 6 factors.
  • The factors of 2023 are outlined with their factor pair partners in the graphic below.

More About the Number 2023:

What do 2023 tiny squares look like?

2023 is the sum of consecutive numbers in five different ways:

And it is the sum of consecutive odd numbers in two ways:

2023 is a palindrome in base 16 because
7(16²) + 14(16) + 7(1) = 2023.

This tweet demonstrates that the prime factors of 2023 have a relationship with the digits of 2023.

I saw this tweet on Twitter and immediately recognized something else cool about 2023.

That might seem like a lot of mathematical mumble jumble, but with a little bit of explanation, it can be understood. And even though I made the problem look scarier because I substituted 2+0+2+3 for 7, some older elementary students who already understand powers, factorials, and/or remainders, will get it. I’m confident you can too.

You could also give the following list of facts to older elementary students and ask them to use it to find the remainder when they divide 823,543 by 5040.

1×5040 =    5040
2×5040 = 10080
3×5040 = 15120
4×5040 = 20160
5×5040 = 25200
6×5040 = 30240
7×5040 = 35280
8×5040 = 40320
9×5040 = 45360

Because its factors, 17 and 289, are hypotenuses of Pythagorean triples, 2023 is also the hypotenuse of some Pythagorean triples:
952-1785-2023 which is 119(8-15-17) , and
1127-1680-2023 which is 7(161-240-289).

Ureczky József also pointed out in the comments of this post, that 2023 is the short leg in SIX Pythagorean triples, and thus
2023² = 2046265² – 2046264²
2023²= 292327² – 292320²
2023² = 120377² – 120360²
2023² = 17255² – 17136²
2023² = 41785² – 41736²
2023² = 7225² – 6936²

One of those triples is a primitive triple. Can you determine which one?

Ureczky József shared one more amazing fact in the comments that I’m replicating here:

2023=9³+8³+7³+6³+5³+4³+3³+2³-1³

More Mathematical Tweets About 2023:

These tweets are more or less in the order I saw them, not in order of mathematical difficulty. I will add more as I see them.


v

1724 Carol of the Ukrainian Bells

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Today’s Puzzle:

As I sit in my warm, peaceful house, I often think about the people of Ukraine whose country has been ravaged by war. Many of them, including children, are also facing a winter with no heat. My heart goes out to them.

My childhood was so unlike theirs. I was able to attend school classes and learn about many different topics without fear of dying. In Junior High School choir class, one of the songs I learned was called Carol of the Bells. I just recently learned from Slate magazine of the song’s Ukrainian roots:

A little over a hundred years ago Mykola Leontovych, a Ukrainian composer, arranged several of his country’s folk songs together in a piece he titled Shchedryk. Tragically, he was murdered by a Russian assassin on January 23, 1921, in the Red Terror, when the Bolsheviks were intent on eliminating Ukrainian leaders, intellectuals, and clergy.

During this time of great unrest, the Ukrainian National Chorus performed Shchedryk around the world and in cities large and small in the United States. One performance was even given at the famed Carnegie Hall on October 5, 1922. The haunting melody was heard by Peter Wilhousky who penned alternate words for it: Hark how the bells, Sweet silver bells,…

Now it is one of our most beloved Christmas carols. I am grateful I learned the words and tune in junior high, although I wish I had learned of its Ukrainian history then as well.

These two bells puzzles are reminiscent of Ukraine’s flag. Long may it wave. Write each number 1 to 10 in the yellow columns and rows so that the given clues are the products of the numbers you write.

Here are the same puzzles if you prefer to use less of your printer ink.

Factors of 1724:

  • 1724 is a composite number.
  • Prime factorization: 1724 = 2 × 2 × 431, which can be written 1724 = 2² × 431.
  • 1724 has at least one exponent greater than 1 in its prime factorization so √1724 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1724 = (√4)(√431) = 2√431.
  • 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 1724 has exactly 6 factors.
  • The factors of 1724 are outlined with their factor pair partners in the graphic below.

More About the Number 1724:

1724 is the difference of two squares:
432² – 430² = 1724.

1724₁₀ = 464₂₀, a palindrome, because
4(20²) + 6(20¹) + 4(20°) = 1724.

1723 A Little Virgács

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Today’s Puzzle:

I haven’t blogged much this year so I guess I deserved a little bit of virgács in my shoes this morning. Mikulás (St. Nick) leaves virgács in the boots of naughty little boys or girls in the wee hours of December 6. Treats are for the good kids. What is virgács? It is small golden spray-painted twigs bound with some pretty red ribbon.  Of course, all children are sometimes naughty and sometimes nice, so they could all expect to get virgács along with their treats in their boots this morning.

You can solve this virgács puzzle by starting with the clues at the top of the grid, finding their factors, and working down the puzzle row by row until you have found all the factors. Each number from 1 to 10 must appear exactly one time in both the first column and the top row.

Factors of 1723:

  • 1723 is a prime number.
  • Prime factorization: 1723 is prime.
  • 1723 has no exponents greater than 1 in its prime factorization, so √1723 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1723 has exactly 2 factors.
  • The factors of 1723 are outlined with their factor pair partners in the graphic below.

How do we know that 1723 is a prime number? If 1723 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1723. Since 1723 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1723 is a prime number.

More About the Number 1723:

1723 and 1721 are twin primes.

1723 is the sum of consecutive numbers:
861 + 862 = 1723.

1723 is the difference of consecutive squares:
862² – 861² = 1723.

1721 A Gift With Multiple Treasures

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Today’s Puzzle:

‘Tis the season of giving, and here’s a gift with multiple treasures inside. 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. After you find the factors, you can complete the puzzle by finding all of the products.

Factors of 1721:

  • 1721 is a prime number.
  • Prime factorization: 1721 is prime.
  • 1721 has no exponents greater than 1 in its prime factorization, so √1721 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1721 has exactly 2 factors.
  • The factors of 1721 are outlined with their factor pair partners in the graphic below.

How do we know that 1721 is a prime number? If 1721 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1721. Since 1721 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1721 is a prime number.

More About the Number 1721:

1721 is the sum of two squares:
40² + 11² = 1721.

Here’s another way we know that 1721 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 11² = 1721 with 40 and 11 having no common prime factors, 1721 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1721. Since 1721 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1721 is a prime number.

1721 and 1723 are twin primes.

1720 Transforming Puzzle-A-Day’s Wizard’s Hat

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Today’s Puzzle:

I recently saw a fun-to-solve puzzle from Puzzle a Day on Twitter:

The puzzle got me thinking, what would happen if the numbers weren’t from 1 to 10, but were from 0 to 9 instead? Would it change the puzzle a little or a lot?

It actually changes the puzzle quite a bit. There are a lot more solutions than for the Puzzle a Day puzzle! And I don’t mean just switching the red line with the green line and/or switching the purple circles at the bottom. How many solutions can you find?

Factors of 1720:

  • 1720 is a composite number.
  • Prime factorization: 1720 = 2 × 2 × 2 × 5 × 43, which can be written 1720 = 2³ × 5 × 43.
  • 1720 has at least one exponent greater than 1 in its prime factorization so √1720 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1720 = (√4)(√430) = 2√430.
  • 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 1720 has exactly 16 factors.
  • The factors of 1720 are outlined with their factor pair partners in the graphic below.

More About the Number 1720:

1720 is the difference of two squares in FOUR different ways:
431² – 429² = 1720,
217² – 213² = 1720,
91² – 81² = 1720, and
53² – 33² = 1720.

 

1717 Factor Fits for Your Valentine

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Today’s Puzzle:

Solve this Valentine-themed Factor Fits puzzle with both logic and heart!

Factors of 1717:

  • 1717 is a composite number.
  • Prime factorization: 1717 = 17 × 101.
  • 1717 has no exponents greater than 1 in its prime factorization, so √1717 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 1717 has exactly 4 factors.
  • The factors of 1717 are outlined with their factor pair partners in the graphic below.

More About the Number 1717:

1717 is the sum of two squares two different ways:
41² + 6² = 1717, and
39² + 14² = 1717.

1717 is the hypotenuse of FOUR Pythagorean triples:
340 1683 1717, which is 17 times (20-99-101),
492 1645 1717, calculated from 2(41)(6), 41² – 6², 41² + 6²,
808 1515 1717, which is (8-15-17) times 101, and
1092 1325 1717, calculated from 2(39)(14), 39² – 14², 39² + 14².

Two Factor Pairs of 1716 Make Sum-Difference!

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Today’s Puzzle:

How many factor pairs does 1716 have? Twelve. The factors in one of those factor pairs add up to 145 and the factors in a different one subtract to 145. If you can find those two factor pairs, then you can solve this puzzle!

Try to solve it all by yourself, but if you need some help, you can scroll down to learn more about the factors of 1716.

A Factor Tree for 1716:

3 is a factor of 1716 because 1 + 7 + 1 + 6 = 15, a multiple of 3.
11 is a factor of 1716 because 1 – 7 + 1 – 6 = -11.

3 ·11 is 33. Here is one of MANY possible factor trees for 1716. This one uses the fact that 33 · 52 = 1716:

Factors of 1716:

  • 1716 is a composite number.
  • Prime factorization: 1716 = 2 × 2 × 3 × 11 × 13, which can be written 1716 = 2² × 3 × 11 × 13.
  • 1716 has at least one exponent greater than 1 in its prime factorization so √1716 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1716 = (√4)(√429) = 2√429.
  • 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 1716 has exactly 24 factors.
  • The factors of 1716 are outlined with their factor pair partners in the graphic below.

Here’s a chart of those same factor pairs but in reverse order with their sums and differences included.

More About the Number 1716:

1716 is the hypotenuse of a Pythagorean triple:
660-1584-1716, which is (5-12-13) times 132.

OEIS.org reminds us that 1716 is in the 6th and 7th columns of the 13th row of Pascal’s Triangle:

1715 A Lot More of a Subtraction Distraction

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Today’s Puzzle:

Last time I published a puzzle with the last clue missing. Leaving out the first or the last clue only makes the puzzle slightly more difficult. What if I left out a clue more in the middle of the puzzle. I gave that some thought and designed today’s puzzle. I soon realized that I had to let you know that the 12 is one of the last eight boxes. There is only one solution. Can you find it?

I posted a solution video for it on Twitter:

Factors of 1715:

  • 1715 is a composite number.
  • Prime factorization: 1715 = 5 × 7 × 7 × 7, which can be written 1715 = 5 × 7³.
  • 1715 has at least one exponent greater than 1 in its prime factorization so √1715 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1715 = (√49)(√35) = 7√35.
  • The exponents in the prime factorization are 1 and 3. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1715 has exactly 8 factors.
  • The factors of 1715 are outlined with their factor pairs in the graphic below.

More About the number 1715:

1715 = 1·7³·1·5. Thank you, OEIS.org for that fun fact!

1715 is the hypotenuse of a Pythagorean triple:
1029-1372-1715, which is (3-4-5) times 343.

1715 = 5·7³.
5·7º = 2² + 1².
5·7¹ cannot be written as the sum of two squares.
5·7² = 14² + 7².
5·7³ cannot be written as the sum of two squares.
5·7⁴ = 98² + 49².

What do you notice? What do you wonder?

1714 Will the Factors in This Puzzle Give You Fits?

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Today’s Puzzle:

12 and 24 have several common factors, but only one of them works in this puzzle. Will it be 2, 3, 4, 6, or 12?

What about 40 and 60’s common factors?

Don’t guess which factor to use. Start elsewhere in the puzzle where there’s only one possible common factor. Then use logic to eliminate some of the factor possibilities for 12, 24 and 40, 60. You will have to think, but it won’t be too difficult.

Factors of 1714:

  • 1714 is a composite number.
  • Prime factorization: 1714 = 2 × 857.
  • 1714 has no exponents greater than 1 in its prime factorization, so √1714 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 1714 has exactly 4 factors.
  • The factors of 1714 are outlined with their factor pair partners in the graphic below.

More About the Number 1714:

1714 is the sum of two squares:
33² + 25² = 1714.

1714 is the hypotenuse of a Pythagorean triple:
464-1650-1714, calculated from 33² – 25², 2(33)(25), 33² + 25².
It is also 2 times (232-825-857).

1713 A Little More of a Subtraction Distraction

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Today’s Puzzle:

It occurred to me that as long as the last box is neither 1 nor 12 that I could leave the clue above it blank. Can you still solve the puzzle?

Factors of 1713:

  • 1713 is a composite number.
  • Prime factorization: 1713 = 3 × 571.
  • 1713 has no exponents greater than 1 in its prime factorization, so √1713 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 1713 has exactly 4 factors.
  • The factors of 1713 are outlined with their factor pair partners in the graphic below.

More About the Number 1713:

1713 is the difference of two squares in two different ways:
857² – 856² = 1713, and
287² – 284² = 1713.

1713 is the sum of two, three, and six consecutive numbers:
856 + 857 = 1713,
570 + 571 + 572 = 1713, and
283 + 284 + 285 + 286 + 287 + 288 = 1713.

Do you see any relationship between those two facts?