Mathematics

1746 Love-Struck with Multiplication

/ ivasallay / Leave a comment

Today’s Puzzle:

If you solve this Valentine’s multiplication table puzzle, you might just become love-struck with multiplication! It’s a level 6 puzzle so you might find most of the clues to be tricky. Be sure to use logic as you place each number from 1 to 10 in both the first column and also in the top row. There is only one solution.

Factors of 1746:

  • 1746 is a composite number.
  • Prime factorization: 1746 = 2 × 3 × 3 × 97, which can be written 1746 = 2 × 3² × 97.
  • 1746 has at least one exponent greater than 1 in its prime factorization so √1746 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1746 = (√9)(√194) = 3√194.
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1746 has exactly 12 factors.
  • The factors of 1746 are outlined with their factor pair partners in the graphic below.

More About the Number 1746:

Why is 1746 the sum of two squares? We can tell by looking at its PRIME factors:
2 is a power of 2,
3² is an even power, and
97 is a prime Pythagorean hypotenuse. (It is prime and one more than a positive multiple of 4).

At least one of its prime factors is a Pythagorean triple hypotenuse, and ALL of the rest of its non-Pythagorean-hypotenuse prime factors are either a power of 2 or combine to make a perfect square. Thus, 1746 is the sum of two squares. What are the two squares?
39² + 15² = 1746.

1746 is the hypotenuse of a Pythagorean triple:
1170-1296-1746, calculated from 2(39)(15), 39² – 15², 39² + 15².
It is also 18 times (65-72-97).

 

1743 Finding Ways to Transform My Heart

/ ivasallay / Leave a comment

Today’s Puzzle:

An equation of a unit circle centered at the origin is x² + y² = 1.

If we change just the “y” part of that equation, we can get a lovely heart just in time for Valentine’s Day. Try it yourself by typing the equations into Desmos.

There are other mathematical equations for a heart, but this is the one I’m exploring in this post.

I was puzzled over how I could transform that heart. Can I make it bigger, or dilate it? Can I slide it away from the origin or translate it? Can I rotate it? Can I reflect it across the x or y-axis?

These are questions I’d like you to explore as well.

Heart Dilation:

In this first graphic, I was able to make my heart bigger. What kind of math let me do that? Also, how did I color the inside of some of the hearts? Look at the equations next to the heart and try to figure it out. The concentric hearts are evenly spaced. Do you recognize a pattern in the numbers that made that happen?

Heart Slide (Translation):

If we changed the center of a circle to (a, -b) instead of the origin, we would slide the whole circle. Here’s how we change the equation of the circle to give it a new center:
(x-a)²+(y+b)² = 1.

Similarly, in the next graphic, I was able to slide my heart away from the origin. How did I do that? Look at the equations to see how.

Heart Rotation:

A circle looks the same no matter how it is rotated, but the same isn’t true for a heart. Look at the equations below. How was I able to rotate my heart around the origin?

Heart Reflection:

Since a heart is symmetric, its reflection across the x-axis doesn’t look that interesting to me. Instead, I created a double heart that I reflected across both the x-axis and the y-axis:

Just for Fun:

Next, I was curious about what would happen if I changed the exponents on the outside of the parenthesis, so I changed a 2 from my original equation to an 8 in a couple of different places as I moved the heart from left to right. How did changing the exponent affect my heart? I found that as long as the exponent stays even, it still looks a little like a heart.

I was also curious about what would happen to my heart if I changed the “2/3” to a different fraction. I used fractions less than one as well as fractions greater than one. For many of my fractions, I used the post number, 1743, as the denominator. As long as the numerator was even and the denominator was odd, the graph still looked mostly like a heart. However, the closer the fraction was to zero, the more it looked like a circle.

Finally, I created this lovely flower using some of what I learned by making these transformations:

And for just a little bit more fun, I created a simple but chaotic-looking animation that I’ve titled Hearts in Motion. Enjoy!

I had so much fun exploring this heart in Desmos. Thank you for allowing me to share my excitement with you. Here’s a different Desmos heart created by a reader and shared with me. Click on it to make the heart beat:

 

Factors of 1743:

  • 1743 is a composite number.
  • Prime factorization: 1743 = 3 × 7 × 83.
  • 1743 has no exponents greater than 1 in its prime factorization, so √1743 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 1743 has exactly 8 factors.
  • The factors of 1743 are outlined with their factor pair partners in the graphic below.

More About the Number 1743:

1743 is the difference of two squares in four different ways:
872² – 871² = 1743,
292² – 289² = 1743,
128² – 121² = 1743, and
52² – 31² = 1743.

1741 What Do You Notice About the Remainders?

/ ivasallay / Leave a comment

Today’s Puzzle:

The table below is a list of the first 280 prime numbers. Below each prime number is its remainder when the prime number is divided by 6. What do you notice about the remainders? What do you wonder?

Did you notice a fun fact starting with the number 1741? I would like to thank OEIS.org for making me aware of it.

Factors of 1741:

  • 1741 is a prime number.
  • Prime factorization: 1741 is prime.
  • 1741 has no exponents greater than 1 in its prime factorization, so √1741 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1741 has exactly 2 factors.
  • The factors of 1741 are outlined with their factor pair partners in the graphic below.

How do we know that 1741 is a prime number? If 1741 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1741. Since 1741 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1741 is a prime number.

More About the Number 1741:

1741 is the sum of two squares:
30² + 29² = 1741.

Here’s another way we know that 1741 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 29² = 1741 with 30 and 29 having no common prime factors, 1741 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1741. Since 1741 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1741 is a prime number.

Since 29 and 30 are consecutive numbers, we have another fun fact:

And 1741 is the hypotenuse of a Pythagorean triple:
59-1740-1741, calculated from 30² – 29², 2(30)(29), 30² + 29².

1741 is also the difference of two consecutive squares:
871² – 870² = 1741.

1737 What Logic Will You Use to Solve This Puzzle?

/ ivasallay / Leave a comment

Today’s Puzzle:

Place each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues. This is a level 6 puzzle so the logic needed to solve the puzzle will be a little more complicated. Good luck!

Factors of 1737:

1 + 7 + 3 + 7 = 18, a number divisible by nine, so 1718 is divisible by 9.

  • 1737 is a composite number.
  • Prime factorization: 1737 = 3 × 3 × 193, which can be written 1737 = 3² × 193.
  • 1737 has at least one exponent greater than 1 in its prime factorization so √1737 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1737 = (√9)(√193) = 3√193.
  • 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 1737 has exactly 6 factors.
  • The factors of 1737 are outlined with their factor pair partners in the graphic below.

More About the Number 1737:

1737 is the difference of two squares in three ways:
869² – 868² = 1737,
291² – 288² = 1737, and
101² – 92² = 1737.

1737 is the sum of consecutive numbers:
868 + 869 = 1737,
578 + 579 +560 = 1737,
287 + 288 + 289 + 290 + 291 + 292 = 1737,
189 + 190 + 191 + 192 + 193 + 194 + 195 + 196 + 197 = 1737, and
88+89+90+91+92+93+94+95+96+97+98+99+100+101+102+103+104+105=1737.

That first sum uses the same numbers as the first difference of two squares. Check out these square equations!

292² – 291² + 290² – 289² + 288² – 287² = 1737.

105² – 104² + 103² – 102² + 101² – 100² + 99² – 98² + 97² – 96² + 95² – 94² + 93² – 92² + 91² – 90² + 89² – 88² = 1737.

(But since 101² – 92² = 1737, we can conclude that all the rest of the numbers added and subtracted will give us zero!)

1737 is also the sum of two squares:
36² + 21² = 1737.

1737 is the hypotenuse of a Pythagorean triple:
855-1512-1737 calculated from 36² – 21², 2(36)(21), 36² + 21².
That triple is also 9 times (95-168-193).

That Pythagorean triple means that
855² + 1512² = 1737².

From OEIS.org, we learn that when we add up these three consecutive squares,
1736² + 1737² + 1738² we get a palindrome, specifically, 9051509. Cool!

1737² = 1736² + 1734² – 1733² + 1732² – 1731² – 1729² + 1728².

1736 Fun With Magic Squares

/ ivasallay / Leave a comment

Today’s Puzzle is a 7×7 Magic Square:

1736 is the sum of seven consecutive numbers:
245 + 246 + 247 + 248 + 249 + 250 + 251 = 1736.

That means 1736 is the magic sum of a 7×7 Magic Square with the numbers from that sum running along the diagonal. Can you write the rest of the consecutive numbers from 224 to 272 to complete the magic square?

If you need some help, trace with your finger while you count from 224 to 272 and see where the numbers fall diagonally on this completed magic square. You should notice that you always place the next number diagonally above to the right unless you can’t, in which case you place the next number directly under your last entry. Sometimes you will have to leave the square and get back on it on the opposite side of the square to maintain the diagonal.

After studying the pattern, try to do it yourself. This excel sheet has a place for you to type the numbers for this magic square and every other magic square discussed in this post. The spreadsheet will even keep a running sum of each column, row, and diagonal as you type in the numbers: 12 Factors 1730-1738.

The current year is also divisible by seven, and consequently
286 + 287 + 288 + 289 + 290 + 291 + 292 = 2023.
Can you make a 7×7 Magic Square with 2023 as the magic sum?

Here is that completed square as well. It follows the same mostly diagonal path as the previous completed magic square:

17×17 Magic Square:

2023 is divisible by 17, and consequently

111 + 112 + 113 + 114 + 115 + 116 + 117 + 118 + 119 + 120 + 121 + 122 + 123 + 124 + 125 + 126 + 127 = 2023.

All that means that 2023 is also the magic sum of a 17×17 Magic Square, but we also have to use some negative numbers to make it. Trace the numbers beginning with -25 and notice the same diagonal pattern in this magic square.

14×14 Magic Square:

It’s a bit trickier, but since 2023 is divisible by seven but not by 14, it is also the magic sum of a 14×14 Magic Square. To make it start off by dividing the 14×14 square into four 7×7 squares. Place the numbers from 47 to 95 in the top left square, the numbers from 96 to 144 in the bottom right square, the numbers from 145 to 193 in the top right square, and the numbers from 194 to 242 in the bottom left square.

Notice that all of the columns show our desired magic sum, but none of the rows or diagonals do. We need to switch some of the numbers to fix that. Switch the shaded areas indicated below with their corresponding darker shaded areas:

And you will successfully create this beautiful magic square where every row, column, and diagonal equals 2023.

Notice that the 14 consecutive numbers that add up to 2023 are all over the square.
138+139+140+141+142+143+144+145+146+147+148+149+150+151 = 2023

Any other magic square for 2023 would be too big and have so many negative numbers.

16×16 Magic Square:

1736 is divisible by 8, but not by 16, so there are 16 consecutive numbers that add up to 1736:

101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 + 109 + 110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 = 1736.

Making a 16×16 Magic Square will be a bit tedious, but so satisfying. Start by placing the numbers from -19 to 236 in order from left to right filling in each 4×4 square before moving onto the next 4×4 square, as illustrated below:

Notice that only the diagonals show the desired sum of 1736.

Next, we want to flip the diagonals of each small 4×4 square, as shown below.

The diagonals still have the correct sum, and look at those sets of four columns or four rows that have equal sums! Now think of the whole square as one big 4×4 Magic Square, and flip its diagonals as shown below:

Oh, it is a thing of beauty, don’t you agree?

Factors of 1736:

  • 1736 is a composite number.
  • Prime factorization: 1736 = 2 × 2 × 2 × 7 × 31, which can be written 1736 = 2³ × 7 × 31.
  • 1736 has at least one exponent greater than 1 in its prime factorization so √1736 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1736 = (√4)(√434) = 2√434.
  • 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 1736 has exactly 16 factors.
  • The factors of 1736 are outlined with their factor pair partners in the graphic below.

More About the Number 1736:

1736 is the difference of two squares in four different ways:
435² – 433² = 1736,
219² – 215² = 1736,
69² – 55² = 1736, and
45² – 17² = 1736.

 

1733 and Its Prime Number Substrings

/ ivasallay / Leave a comment

Today’s Puzzle:

Use logic to write each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.

Factors of 1733:

  • 1733 is a prime number.
  • Prime factorization: 1733 is prime.
  • 1733 has no exponents greater than 1 in its prime factorization, so √1733 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1733 has exactly 2 factors.
  • The factors of 1733 are outlined with their factor pair partners in the graphic below.

How do we know that 1733 is a prime number? If 1733 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1733. Since 1733 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1733 is a prime number.

More About the Number 1733:

1733 is the sum of two squares:
38² + 17² = 1733.

1733 is the hypotenuse of a Pythagorean triple:
1155-1292-1733 calculated from 38² – 17², 2(38)(17), 38² + 17².

Here’s another way we know that 1733 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 17² = 1733 with 38 and 17 having no common prime factors, 1733 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1733. Since 1733 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1733 is a prime number.

1733 is also the difference of two squares:
867² – 866² = 1733.
That means 1733 is also the short leg of the Pythagorean triple calculated from
867² – 866², 2(867)(866), 867² + 866².

1733 is a palindrome in three bases:
It’s 2101012 in base 3 because
2(3⁶) +1(3⁵) + 0(3⁴) + 1(3³) + 0(3²) +1(3¹) + 2(3⁰) = 1733,
It’s 565 in base 18 because 5(18²) + 6(18) + 5(1) = 1733, and
it’s 4F4 in base 19 because 4(19²) + 15(19) + 4(1) = 1733.

The next fact I learned from Twitter:

What prime numbers can be made with the digits of 1733?
3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 313, 317, 331, 337, 373, 733, 1733.

The numbers in red are the 6 prime numbers that are substrings of 1733. I made a gif showing those 6 primes:

1733 Smallest Prime with 6 Prime Substrings

make science GIFs like this at MakeaGif
1733 is quite a fascinating number!

1729 Not a Dull Number

/ ivasallay / Leave a comment

Today’s Puzzle:

A little more than a hundred years ago near Cambridge University G. H. Hardy took a taxi to visit his young friend and fellow mathematician, Srinivasa Ramanujan, in the hospital. Hardy couldn’t think of anything interesting about his taxi number, 1729, and remarked to Ramanujan that it appeared to be a rather dull number. But even the reason for his hospitalization could not prevent Ramanujan’s genius from shining through. He immediately recognized 1729’s unique and very interesting attribute: it is the SMALLEST number that can be written as the sum of two cubes in two different ways! Indeed,
12³ + 1³ = 1729, and
10³ + 9³ = 1729.

Today’s puzzle looks a little bit like a modern-day American taxi cab with the clues 17 and 29 at the top of the cab. The table below the puzzle contains all the Pythagorean triples with hypotenuses less than 100 sorted by legs and by hypotenuses. Use the table and logic to write the missing sides of the triangles in the puzzle. The right angle on each triangle is the only one that is marked. Obviously, none of the triangles are drawn to scale. No Pythagorean triple will appear more than once in the puzzle.

Sorted TriplesHere’s the same puzzle without all the added color:

Print the puzzles or type the solutions in this excel file: 10 Factors 1721-1729.

What taxi cab might Hardy have tried to catch next? He might have had to wait a long time for it, 4104.
16³ + 2³ = 4104, and
15³ + 9³ = 4104.

Factors of 1729:

  • 1729 is a composite number.
  • Prime factorization: 1729 = 7 × 13 × 19.
  • 1729 has no exponents greater than 1 in its prime factorization, so √1729 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 1729 has exactly 8 factors.
  • The factors of 1729 are outlined with their factor pair partners in the graphic below.

More About the Number 1729:

Did you notice these cool-looking factor pairs?
19 · 91 = 1729.
13 · 133 = 1729.

1729 is the hypotenuse of a Pythagorean triple:
665-1596-1729 which is 133 times (5-12-13).

1729 is the difference of two squares in four different ways:
865² – 864² = 1729,
127² – 120² = 1729,
73² – 60² = 1729, and
55² – 36² = 1729.

1729 is also a pseudoprime number! For example, even though it is a composite number, it passes the quick prime number test I first wrote about in 341 is the smallest composite number that gives a false positive for this Quick Prime Number Test:

It is known as a Carmichael number because back in 1909, Robert D. Carmichael, an American mathematician, showed that it passed even more prime number tests than this one.

1727 A Happy Birthday Cake for Jo Morgan

/ ivasallay / 2 Comments

Today’s Puzzle:

Today is Jo Morgan’s birthday, so I made her a cake!

Use logic to find the factors in the cake. There is only one solution. Yesterday’s puzzle was pretty easy at least to get started. This one won’t be. You’ll have to use logic just to figure out which set of numbers belongs in the top row, 1-12 OR 13-24, and which set belongs in the first column. Not only that but both 126 and 180 appear THREE times in the puzzle! Have a party figuring it out!

Factors of 1727:

1 – 7 + 2 – 7 = -11, a multiple of 11, so 1727 is divisible by 11.

  • 1727 is a composite number.
  • Prime factorization: 1727 = 11 × 157.
  • 1727 has no exponents greater than 1 in its prime factorization, so √1727 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 1727 has exactly 4 factors.
  • The factors of 1727 are outlined with their factor pair partners in the graphic below.

More About the Number 1727:

1727 is the hypotenuse of a Pythagorean triple:
935-1452-1727, which is 11 times (85-132-157).

From OEIS we learn that
1727 = 12³ – 1³, and
7271 = 20³ – 9³.
Thus 1727 forward and backward is the difference of two cubes!

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

1722 Mikulás, Fill My Boot With Goodies!

/ ivasallay / Leave a comment

Today’s Puzzle:

Today is December 5, so children in Hungary and some other countries will shine their boots and put them out for Mikulás (Saint Nick) to fill with candies. Can you solve this boot puzzle? Write the numbers 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.

Here is a song children will sing in Hungary today. You are already familiar with the tune. This particular video includes subtitles.

Here are the lyrics in Hungarian and Google’s translation into English:

Hull a pelyhes fehér hó, jöjj el kedves Télapó!
Minden gyermek várva vár, vidám ének hangja száll.
Van zsákodban minden jó, piros alma, mogyoró,
Jöjj el hozzánk, várunk rád, kedves öreg Télapó.

Nagy szakállú Télapó jó gyermek barátja.
Cukrot,diót, mogyorót rejteget a zsákja.
Amerre jár reggelig kis cipőcske megtelik,
megtölti a Télapó, ha üresen látja!

The fluffy white snow is falling, come dear Santa!
Every child waits expectantly, the sound of happy singing is heard.
You have everything good in your bag, red apples, hazelnuts,
Come to us, we are waiting for you, dear old Santa Claus.

Santa Claus with a big beard is a good friend of children.
His sack hides sugar, walnuts and hazelnuts.
Wherever you go, your little shoe will be full by morning,
Santa will fill it if he sees it empty!

Factors of 1722:

  • 1722 is a composite number.
  • Prime factorization: 1722 = 2 × 3 × 7 × 41.
  • 1722 has no exponents greater than 1 in its prime factorization, so √1722 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1722 has exactly 16 factors.
  • The factors of 1722 are outlined with their factor pair partners in the graphic below.

More About the Number 1722:

Since 1722 = 41·42, we know that 1722 is the sum of the first 41 EVEN numbers. It is two times the 41st triangular number.