Prime factorization

1736 Fun With Magic Squares

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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.

 

1735 What Amazing Mathematical Pattern I Noticed Today!

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Today’s Puzzle:

Will the common factor of 60 and 10 be 5 or will it be 10? Don’t guess. There is a better place to start this puzzle. After using logic to find some of the other factors, you will know if you should use 5 or 10. Don’t let me trip you up!

Factors of 1735:

  • 1735 is a composite number.
  • Prime factorization: 1735 = 5 × 347.
  • 1735 has no exponents greater than 1 in its prime factorization, so √1735 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 1735 has exactly 4 factors.
  • The factors of 1735 are outlined with their factor pair partners in the graphic below.

More About the Number 1735 and an Amazing Pattern:

1735 is the hypotenuse of a Pythagorean triple:
1041-1388-1735, which is (3-4-5) times 347.

1735 is 567 in base 18 because
5(18²) + 6(18) + 7(1) = 1735.

I know neither of those facts is amazing, but read on:

1735 is the sum of two, five, and ten consecutive positive numbers:
867 + 868 = 1735,
345 + 346 + 347 + 348 + 349 = 1735, and
169 + 170 + 171 + 172 + 173 + 174 + 175 + 176 + 177 + 178 = 1735.

1735 is the difference of two squares in two different ways:
868² – 867² = 1735, and
176² – 171² = 1735.

That first difference of two squares used the two consecutive numbers I listed above in red. Since mathematics is the study of patterns, I wondered if there would also be a relationship of squares with the ten consecutive numbers in blue. I typed into my calculator:
178² – 177² + 176² – 175² + 174² – 173² + 172² – 171² + 170² – 169² =

Guess what the answer was? If you guessed 1735, you would be right! Oh, mathematics is certainly filled with amazing patterns! I’m thrilled that I spotted this one! It appears something similar can be written whenever a number is the sum of an even amount of consecutive numbers! I find that amazing!

I also noticed that the second difference of two squares noted above is contained in the string. I underlined those two squares. Since the difference of those squares by themselves equals the whole string, it follows that the rest of those squares added and subtracted equal zero. That means the sum of the rest of the positive terms must equal the sum of the rest of the negative terms:
178² + 174² + 172² + 170² = 120144 = 177² + 175² + 173² + 169².
I’ve never found a string of four squares equal to another string of four squares before, so naturally, I’m excited! I’m anxious to apply this pattern to other numbers that are the sum of ten consecutive numbers. For example, the first ten numbers add up to 55.
10² – 9² + 8² – 7² + 6² – 5² + 4² – 3² + 2² – 1² =55. And since 8² – 3² = 55, it follows that
10²  + 6² + 4² + 2² = 156 = 9² + 7² + 5²+ 1².
It is no longer “impossible” for me to find these equal sums! Here’s another that connects this year with other recent years:

We can all hope this equation doesn’t predict the future. Ha ha! I’m confident it is NOT a bad omen!

Both17 and 34 are Factors of 1734 and Help It Make Sum-Difference

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Today’s Puzzle:

1734 has only 6 factor pairs, but two of those pairs will solve a Sum-Difference puzzle.

Let’s start with the first puzzle below. Can you find a factor pair of 6 that adds up to 5 and another one that subtracts to 5? If you can, then you have solved the first puzzle. The second puzzle is really just a multiple of the first puzzle. 5 × what = 85? Multiply the numbers you write in the first puzzle by that same number, and you will have the numbers needed to solve the second puzzle. You can also look at the factor pairs in the next section to help you find the needed factors that sum or subtract to 85.

Factors of 1734:

Because 17×2=34, their concatenation, 1734, is divisible by three. All of the following concatenated numbers are also divisible by three: 12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, 1530, 1632, 1734, 1836, 1938, and so forth.

  • 1734 is a composite number.
  • Prime factorization: 1734 = 2 × 3 × 17 × 17, which can be written 1734 = 2 × 3 × 17².
  • 1734 has at least one exponent greater than 1 in its prime factorization so √1734 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1734 = (√289)(√6) = 17√6.
  • 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 1734 has exactly 12 factors.
  • The factors of 1734 are outlined with their factor pair partners in the graphic below.

More About the Number 1734:

Did you notice that both 17 and 34 are factors of 1734?

From OEIS.org we learn that 1734 has three factors that have 8 as one of their digits, 867, 578, and 289. What do you get when you add those factors together?

1734 is the hypotenuse of two Pythagorean triples:
816-1530-1734, which is (8-15-17) times 102. and
966-1440-1734, which is 6 times (161-240-289).

1734 looks interesting in some other bases:
It’s 600 in base 17 because 6(17²) = 1734, and
it’s 369 in base 23 because 3(23²) + 6(23) + 9(1) = 1734.

What is the greatest common factor of 1734 and 2023? What is the least common multiple? The easiest way to know is to look at their prime factorizations.
1734 = 2·3·17²,
2023 = 7·17².
The greatest common factor is what they have in common: 17² = 289.
The least common multiple is 2·3·7·17² = 12138.

1732 Is a Thousand Times More Than…

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Today’s Puzzle:

If you know the common factor of 50 and 35 that will put only numbers from 1 to 12 in the first column and the top row of this multiplication table puzzle, then you will have completed the first step in solving the puzzle. Afterward, just work from the top of the puzzle to the bottom filling in factors as you go. That’s how you solve these level-three puzzles.

Factors of 1732:

  • 1732 is a composite number.
  • Prime factorization: 1732 = 2 × 2 × 433, which can be written 1732 = 2² × 433.
  • 1732 has at least one exponent greater than 1 in its prime factorization so √1732 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1732 = (√4)(√433) = 2√433.
  • 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 1732 has exactly 6 factors.
  • The factors of 1732 are outlined with their factor pair partners in the graphic below.

More About the Number 1732:

1732 is the sum of two squares:
34² + 24² = 1732.

1732 is the hypotenuse of a Pythagorean triple:
580-1632-1732, calculated from 34² – 24², 2(34)(24), 34² + 24².
It is also 4 times (145-408-433).

1732 is a palindrome in base 15 because
7(15²) + 10(15) + 7(1) = 1732.

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentine’s day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

That’s how I remember those values, but this tweet reminded me that 1732 is a thousand times more than the square root of three rounded to three decimal places. It also makes a reference to the square root of two.

1731: The Sum of the Squares of Three Consecutive Prime Numbers

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Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that the given clues are the products of the numbers you write.

Factors of 1731:

  • 1731 is a composite number.
  • Prime factorization: 1731 = 3 × 577.
  • 1731 has no exponents greater than 1 in its prime factorization, so √1731 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 1731 has exactly 4 factors.
  • The factors of 1731 are outlined with their factor pair partners in the graphic below.

More About the Number 1731:

1731 is the hypotenuse of a Pythagorean triple:
144-1725-1731 which is 3 times (48-575-577).

OEIS.org informs us that 1731 is the sum of the squares of three consecutive prime numbers. Let’s find those three prime numbers. Since √(1731/3) rounds to 24, I’m guessing the middle prime number is 23. The prime numbers occurring before and after it are 19 and 29.

Is 19² + 23² + 29² = 1731? Yes, it is!

Here are some other ways that 1731 is the sum of three squares:
41² + 7² + 1² = 1731,
41² + 5² + 5² = 1731,
37² + 19² + 1² = 1731, and
29² + 29² + 7² = 1731.

How Is 1730 the Sum of Consecutive Squares?

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Today’s Puzzle:

Write all the numbers 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 1730:

  • 1730 is a composite number.
  • Prime factorization: 1730 = 2 × 5 × 173.
  • 1730 has no exponents greater than 1 in its prime factorization, so √1730 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 1730 has exactly 8 factors.
  • The factors of 1730 are outlined with their factor pair partners in the graphic below.

More about the number 1730:

1730 is the sum of two squares in two different ways:
41² + 7² = 1730, and
37² + 19² = 1730.

1730 is the hypotenuse of FOUR Pythagorean triples:
520 1650 1730 which is 10 times (52-165-173)
574 1632 1730 calculated from 2(41)( 7), 41² – 7², 41² + 7²,
1008 1406 1730 calculated from 37² – 19², 2(37)(19), 37² + 19², and
1038 1384 1730 which is 346 times (3-4-5).

Finally, OEIS.org informs us that 1730 is the sum of consecutive squares in two different ways. What are those two ways? I figured it out. Can you?

Here’s a hint: It is the sum of three consecutive squares as well as twelve consecutive squares. That means √(1730/3) rounded is included in one sum and √(1730/12) rounded is included in the other. The solution can be found in the comments. Have fun finding them yourself though!

1729 Not a Dull Number

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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.

1728 Is a Perfect Cube

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Today’s Puzzle:

I think this puzzle is a little bit harder than the previous two puzzles, so you might want to solve those two first. You will need to write all the numbers from 1 to 24 where the factors go, but keep the numbers from 1 to 12 together and the numbers from 13 to 24 together. There is only one solution. The possible factors are written to the right of the puzzle. Good luck!

You can solve the puzzle by eliminating factors that must be used for other numbers. For example, 198 cannot be 9×22 because 189 must be 9×21, and the puzzle can only have one nine. Don’t write the factors on the multiplication table until after you know if the top row has the numbers 1 – 12 or if it has the numbers 13 – 24. The first column will then have the other set of numbers.

Factors of 1728:

Let’s find out the factoring information of the puzzle number:

  • 1728 is a composite number and a perfect cube.
  • Prime factorization: 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be written 1728 = 2⁶ × 3³.
  • 1728 has at least one exponent greater than 1 in its prime factorization so √1728 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1728 = (√576)(√3) = 24√3.
  • The exponents in the prime factorization are 6 and 3. Adding one to each exponent and multiplying we get (6 + 1)(3 + 1) = 7 × 4 = 28. Therefore 1728 has exactly 28 factors.
  • The factors of 1728 are outlined with their factor pairs in the graphic below.

More About the Number 1728:

1728 is the 3rd smallest number with exactly 28 factors.

There are 1728 cubic inches in one cubic foot.

12³₁₀ = 10³₁₂ (The cube above is 12 cubed in base 10, and it’s 10 cubed in base 12.)

1728 can be written as the sum of powers in some interesting ways:

2¹⁰ + 2⁹ + 2⁷ + 2⁶ = 1024 + 512 + 128 + 64 = 1728.
1(4⁵) + 2(4⁴) + 3(4³) = 1728.
1(6⁴) + 2(6³) = 1728.
3(8³) + 3(8²) = 3(8³ + 8²) = 1728.

11³ = 1331, and 1(11³) + 3(11²) + 3(11¹) + 1(11⁰) = 1728.

Here’s another connection with the number 11.
3(11²) = 363, and
1728 is palindrome 363 in base 23 because
3(23²) + 6(23¹) + 3(23⁰) = 1728.

1728 is 300 in base 24 because 3(24²) = 1728.

I’ve enjoyed researching the number 1728, and I hope you’ve learned some new and interesting fact about it today.

1726 Find the Factors 1-12 AND 13-24

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Today’s Puzzle:

Here’s a puzzle I made to start off 2023. Either all the numbers from 1 to 12 will go in the top row OR they will all go in the first column. All the numbers from 13 to 24 will go in the remaining area. The possible factors are written on the side of the puzzle. Can you find all the factors? There is only one solution.

Factors of 1726:

  • 1726 is a composite number.
  • Prime factorization: 1726 = 2 × 863.
  • 1726 has no exponents greater than 1 in its prime factorization, so √1726 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 1726 has exactly 4 factors.
  • The factors of 1726 are outlined with their factor pair partners in the graphic below.

More About the Number 1726:

1726 is the sum of four consecutive numbers from 430 to 433.

1725 A Toast to You on This Last Day of the Year

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Today’s Puzzle:

I haven’t posted as much this past year as I ought, but thank you for bearing with me. Here is a toast to you, faithful reader! Here’s hoping for a great new year for all of us! 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 factors you write.

Factors of 1725:

  • 1725 is a composite number.
  • Prime factorization: 1725 = 3 × 5 × 5 × 23, which can be written 1725 = 3 × 5² × 23.
  • 1725 has at least one exponent greater than 1 in its prime factorization so √1725 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1725 = (√25)(√69) = 5√69.
  • 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 1725 has exactly 12 factors.
  • The factors of 1725 are outlined with their factor pair partners in the graphic below.

More About the Number 1725:

1725 is the difference of two squares in six different ways:

863² – 862² = 1725,
289² – 286² = 1725,
175² – 170² = 1725,
65² – 50² = 1725,
49² – 26² = 1725, and
47² – 22² = 1725.