Here are 1756 tiny squares. Solve for n and you will know which centered pentagonal number 1756 is. You could also count the pentagons from the center outward.
What triangular number multiplied by 5 is one less than 1756?
How do you pronounce pentagonal? Here’s a quick video on its correct pronunciation:
Confession: I had been mispronouncing all of those terms, but not anymore!
Factors of 1756:
1756 is a composite number.
Prime factorization: 1756 = 2 × 2 × 439, which can be written 1756 = 2² × 439.
1756 has at least one exponent greater than 1 in its prime factorization so √1756 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1756 = (√4)(√439) = 2√439.
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 1756 has exactly 6 factors.
The factors of 1756 are outlined with their factor pair partners in the graphic below.
More About the Number 1756:
The centered pentagon above doesn’t really look like π, so when does 1756 look a little bit like π? Pi day in America is next week, so this is a time-sensitive question.
Actually,1756 looks a little like 10π in base 24. You see 1756₁₀ is the same as 314₂₄.
Why? Because 3(24²) + 1(24) + 4(1) = 1756.
1756₁₀ also is the same as 1024₁₂.
Why? Because 1(12³) + 0(12²) + 2(12) + 4(1) = 1756.
I think that’s cool because 2¹º = 1024.
1756 is the difference of two squares:
440² – 438² = 1456.
Charles Sanders got me thinking about prime decades and how to find them. What a great puzzle for me to work on!
A prime decade is a set of four prime numbers that look the same except for their last digit.
The first decade has 4 prime numbers, 2, 3, 5, 7, but it is NOT a prime decade because 1 and 9 are not prime numbers. 5, 7, 11, 13 is the only prime quadruplet that is not also a prime decade. The first prime decade is 11, 13, 17, 19.
Could 1751, 1753, 1757, and 1759 form a prime decade? Since 1755 is divisible by 15, none of those other four odd numbers can be divisible by 3 or 5, so it is worth looking into if they are all prime numbers. We don’t have to test all four numbers, however, we only need to test 1759 using modular arithmetic. If we note the remainder when we divide 1759 by each prime number from 7 to 41, we will know. (We only have to go to 41 because √1759 < 43, so we don’t need any prime number divisor greater than or equal to 43.)
Will any remainder be a red flag? Yes, if any remainder is 0, 2, 6, or 8, then one of those four odd numbers won’t be prime. Also, if we divide 1759 by 7 and get a remainder of 1, then 1758 and 1751 will both be divisible by 7, and we won’t have a prime decade.
I did the divisions and everything looked great except
1759 ÷ 7 has a remainder of 2, so 1759 – 2 or 1757 is divisible by 7, and
1759 ÷ 17 has a remainder of 8, so 1759 – 8 or 1751 is divisible by 17.
Thus, sadly, the decade ending in 1759 is NOT a prime decade. 🙁
In general, to find prime decades we only need to test the last number of the decades and we only need to test decades if the last number minus 4 is divisible by 15. I made a chart showing only those last numbers, and they appear in the first column. The top two rows are the prime number divisors and their squares. We will divide the last number in the decades by the prime numbers whose square is less than that last number. The remainders in pink are red flags. That would be a 1 in the 7’s column, and 0, 2, 6, or 8 in any column. Any of those would disqualify a decade from being a prime decade. 🙁
The green last numbers (in the first column) show the last number of actual prime decades. 🙂
The yellow last numbers have only ONE divisor that disqualifies the decade. 😐
What are the prime decades in this first millennia?
As you can see, decades ending in 19, 109, 199, and 829 are all prime decades. This chart can also be useful in finding prime triplets because at least two of the prime numbers in a prime triplet whose first prime is greater than 10 must come from a decade whose middle number is divisible by 15. If a decade has two prime triplets in it that meet that qualification, it is also a prime decade.
In the next graphic, we see that the decades ending in 1489 and 1879 are prime decades. Here we also test the decade in which our post number, 1755, appears. We see 1759 has remainders of 2 and 8, confirming that 1757 and 1751 are not prime numbers.
There is only one prime decade, 2081, 2083, 2087, 2089, in the next set of decades, and only one other decade that even came close to being a second one.
There are two prime decades in the next group and several close calls:
None in the next group, but there are several decades that ALMOST qualified.
There’s a prime decade is in this group as well as a handful of decades that almost qualified.
In this next group, there are only two decades that even came close to qualifying.
The next two millennia have no prime decades but they do have several decades that came close to qualifying.
Finally, we get another prime decade in this next group! (We would have had one right away if only 9047 wasn’t divisible by 83.)
I could go on literally forever, but I’ll stop now. Instead, I’ll write a little more about the post number, 1755.
Factors of 1755:
1755 is a composite number.
Prime factorization: 1755 = 3 × 3 × 3 × 5 × 13, which can be written 1755 = 3³ × 5 × 13.
1755 has at least one exponent greater than 1 in its prime factorization so √1755 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1755 = (√9)(√195) = 3√195.
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 1755 has exactly 16 factors.
The factors of 1755 are outlined with their factor pair partners in the graphic below.
More About the Number 1755:
1755 is the hypotenuse of FOUR Pythagorean triples:
432-1701-1755, which is 27 times (16-63-65)
675-1620-1755, which is (5-12-13) times 135,
891-1512-1755, which is 27 times (33-56-65), and
1053-1404-1755, which is (3-4-5) times 351.
Can you write each number from 1 to 12 in the first column and again in the top row so that the products of the numbers you write are the given clues?
Factors of 1754:
1754 is a composite number.
Prime factorization: 1754 = 2 × 877.
1754 has no exponents greater than 1 in its prime factorization, so √1754 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 1754 has exactly 4 factors.
The factors of 1754 are outlined with their factor pair partners in the graphic below.
More About the Number 1754:
1754 is the sum of two squares:
35² + 23² = 1754.
1754 is the hypotenuse of a Pythagorean triple:
696-1610-1754, calculated from 35² – 23², 2(35)(23), 35² + 23².
It is also 2 times (348-805-877).
With so much snow all around it is easy to think about summer. Last summer my family and I stayed at a magnificent Airbnb. There was more than enough room for my husband and me, our children and their spouses, and our grandchildren (32 people in total).
The owners of the Airbnb wrote that the best views come from homes with steep driveways. I will share some of those views later in the post. But just how steep was the driveway? I took a picture and added a grid so you can calculate the slope. Notice that the driveway zigzags. You can estimate where the front tire of the van parked at the top of the driveway hits the pavement to calculate the slope of the top part of the driveway.
Here’s the top half of the driveway:
The beautiful view from the top of the driveway:
The next three pictures are gorgeous views from inside the house or from a balcony:
The views definitely made the steep driveway worth it!
Since this is my 1753rd post, I’ll write a little about that number.
Factors of 1753:
1753 is a prime number.
Prime factorization: 1753 is prime.
1753 has no exponents greater than 1 in its prime factorization, so √1753 cannot be simplified.
The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1753 has exactly 2 factors.
The factors of 1753 are outlined with their factor pair partners in the graphic below.
How do we know that 1753 is a prime number? If 1753 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1753. Since 1753 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1753 is a prime number.
More About the Number 1753:
1753 is a powerful sum:
2¹º + 3⁶ = 1753.
Since both of those exponents are even, 1753 is also the sum of two squares:
32² + 27² = 1753.
1753 is the hypotenuse of a Pythagorean triple:
295-1728-1753, calculated from 32² – 27², 2(32)(27), 32² + 27².
Here’s another way we know that 1753 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 27² = 1753 with 32 and 27 having no common prime factors, 1753 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1753. Since 1753 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1753 is a prime number.
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).
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.
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.
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.
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.
(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!
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:
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:
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.
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.
