Prime factorization

1749 Happy Valentine’s Day!

/ ivasallay / 4 Comments

Today’s Puzzle:

Here’s one more Valentine’s puzzle for you to do. Enjoy!

Factors of 1749:

1749 is divisible by 3 because 7 is half of 14 so any 3-digit number made from the digits 714 in any order is divisible by 3, and of course, 9 is divisible by 3.

1749 is divisible by 11 because 1 – 7 + 4 – 9 = -11, a multiple of 11.

  • 1749 is a composite number.
  • Prime factorization: 1749 = 3 × 11 × 53.
  • 1749 has no exponents greater than 1 in its prime factorization, so √1749 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 1749 has exactly 8 factors.
  • The factors of 1749 are outlined with their factor pair partners in the graphic below.

More About the Number 1749:

1749 is the hypotenuse of a Pythagorean triple:
924-1485-1749 which is 33 times (28-45-53).

1749 is the difference of two squares in FOUR different ways:
875² – 874² = 1749,
293² – 290² = 1749,
85² – 74² = 1749, and
43² – 10² = 1749.

1748 A Factor Fits Valentine’s Puzzle

/ ivasallay / Leave a comment

Today’s Puzzle:

Can you find the one and only solution to this Factor Fits Valentine puzzle or will it give you fits?

Factors of 1748:

  • 1748 is a composite number.
  • Prime factorization: 1748 = 2 × 2 × 19 × 23, which can be written 1748 = 2² × 19 × 23.
  • 1748 has at least one exponent greater than 1 in its prime factorization so √1748 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1748 = (√4)(√437) = 2√437.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1748 has exactly 12 factors.
  • The factors of 1748 are outlined with their factor pair partners in the graphic below.

More About the Number 1748:

1748 is the difference of two squares in two ways:
438² – 436² = 1748, and
42² – 4² = 1748.
That means we are only 16 numbers away from the next perfect square!

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).

 

1745 How Thorny Is This Rose?

/ ivasallay / Leave a comment

Today’s Puzzle:

How thorny is this Valentine’s rose? That depends on if you use logic to solve it! It is a Level 5 puzzle, so some of the clues are meant to be tricky. Nevertheless, you can solve it! Just place all 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. There is only one solution.

Here’s the same puzzle minus the color:

Factors of 1745:

  • 1745 is a composite number.
  • Prime factorization: 1745 = 5 × 349.
  • 1745 has no exponents greater than 1 in its prime factorization, so √1745 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 1745 has exactly 4 factors.
  • The factors of 1745 are outlined with their factor pair partners in the graphic below.

More About the Number 1745:

1745 is the difference of two squares in two different ways:
873² – 872² = 1745, and
177² – 172² = 1745.

1745 is also the sum of two squares in two different ways:
41² + 8² = 1745, and
31² + 28² = 1745.

1745 is the hypotenuse of FOUR Pythagorean triples:
177-1736-1745, calculated from 31² – 28², 2(31)(28), 31² + 28²,
656-1617-1745, calculated from 2(41)(8), 41² – 8², 41² + 8²,
900-1495-1745, which is 5 times (180-299-349)
1047-1396-1745, which is (3-4-5) times 349.

1744 XOXO Kisses or Multiplication? XOXO

/ ivasallay / Leave a comment

Today’s Puzzle:

When I was a little girl, my mother told me that OXOXOX stands for Hugs and Kisses. That would mean that O means “hug” and X means “kiss”, but I think X also means multiplication. It’s almost Valentine’s Day, so in today’s puzzle, it can mean either one.

This is a Level-4 puzzle so the clues don’t come in any particular order. Use logic to place all the numbers 1 to 10 in the first column and in the top row.

Factors of 1744:

  • 1744 is a composite number.
  • Prime factorization: 1744 = 2 × 2 × 2 × 2 × 109, which can be written 1744 = 2⁴ × 109.
  • 1744 has at least one exponent greater than 1 in its prime factorization so √1744 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1744 = (√16)(√109) = 4√109.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1744 has exactly 10 factors.
  • The factors of 1744 are outlined with their factor pair partners in the graphic below.

More About the Number 1744:

1744 is the difference of two squares in three ways:
437² – 435² =1744,
220² – 216² =1744, and
113² – 105² =1744.

1744 is also the sum of two squares:
40² + 12² = 1744.

1744 is the hypotenuse of a Pythagorean triple:
960-1456-1744, calculated from 2(40)(12), 40² – 12², 40² + 12².
It is also 16 times (60-91-109).

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.

1742 Just How Sharp Do You Think Cupid’s Arrow Is?

/ ivasallay / Leave a comment

Today’s Puzzle:

This puzzle looks a little like an arrow that Cupid might be aiming your way. I’m not sure how sharp it is, but here’s hoping that it’s not the least bit painful.

Since this is a level-3 puzzle, start with the clues at the top of the puzzle and work your way down cell by cell until you have found all the factors. Each number from 1 to 10 can only appear one time in the first column and one time in the top row. There is only one solution.

Factors of 1742:

  • 1742 is a composite number.
  • Prime factorization: 1742 = 2 × 13 × 67.
  • 1742 has no exponents greater than 1 in its prime factorization, so √1742 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 1742 has exactly 8 factors.
  • The factors of 1742 are outlined with their factor pair partners in the graphic below.

More About the Number 1742:

1742 is the hypotenuse of a Pythagorean triple:
670-1608-1742 which is (5-12-13) times 134.

1742 looks interesting in some other bases:
It’s 23432 in base 5 because 2(5⁴) + 3(5³) + 4(5²) + 3(5) + 2(1) = 1742,
2345 in base 9 because 2(9³) + 3(9²) + 4(9) + 5(1) = 1742, and
repdigit 222 in base 29 because 2(29² + 29 + 1) = 1742.

1740 I Think You’re Going to L♥ve This Multiplication Puzzle

/ ivasallay / Leave a comment

Today’s Puzzle:

Here’s a heart-shaped level 2 puzzle for you to do. Just place all the numbers from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues.

Here’s the same puzzle in black and white:

Factors of 1740:

  • 1740 is a composite number.
  • Prime factorization: 1740 = 2 × 2 × 3 × 5 × 29, which can be written 1740 = 2² × 3 × 5 × 29.
  • 1740 has at least one exponent greater than 1 in its prime factorization so √1740 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1740 = (√4)(√435) = 2√435.
  • 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 1740 has exactly 24 factors.
  • The factors of 1740 are outlined with their factor pair partners in the graphic below.

More About the Number 1740:

1740 is the hypotenuse of FOUR Pythagorean triples:
204-1728-1740, which is 3 times (68-576-580),
288-1716-1740, which is 3 times (96-572-580),
1044-1392-1740, which is (3-4-5) times 348, and
1200-1260-1740, which is (20-21-29) times 60.

1740 is palindrome 606 in base 17 because
6(17²) + 0(17) + 6(1) = 1740.

1740 is the difference of two squares in four different ways:
436² – 434² = 1740,
148² – 142² = 1740,
92² – 82² = 1740, and
44² – 14² = 1740.

1739 A Gift for a Valentine

/ ivasallay / Leave a comment

Today’s Puzzle:

Here’s a level 1 Valentine’s gift puzzle to start off the month of February. Place the numbers 1 to 10 in both the first column and the top row so that those numbers are the factors of the given number clues.

And here’s a heartless copy of the same puzzle that may be more printer-friendly.

Factors of 1739:

Is 1739 a prime number? It isn’t divisible by 2 or 5.
1 + 7 + 3 + 9 = 20, so it isn’t divisible by 3.
1 – 7 + 3 – 9 = -12, so it isn’t divisible by 11.
√1739 is a little more than 41. Should I try dividing 1739 by every other prime number less than 41? I don’t know divisibility tricks for most of those prime numbers!

Here’s a shortcut I first wrote about in 341 is the smallest composite number that gives a false positive for this Quick Prime Number Test: First I type into my computer’s scientific calculator:
2  xʸ  1739  Mod  1739  =

If the answer on the calculator is anything other than “2”, then it can’t be a prime number. If it is “2”, it very likely is prime. As you can see by the screenshot of my calculator, 1739 is not prime!

  • 1739 is a composite number.
  • Prime factorization: 1739 = 37 × 47.
  • 1739 has no exponents greater than 1 in its prime factorization, so √1739 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 1739 has exactly 4 factors.
  • The factors of 1739 are outlined with their factor pair partners in the graphic below.

More About the number 1739:

1739 is the difference of two squares in two ways:
870² – 869² = 1739, and
42² – 5² = 1739.

That means 1739 is only 25 numbers away from the next perfect square!

1738 Little Surprises Around Every Turn

/ ivasallay / Leave a comment

Today’s Puzzle:

Is this mystery-level puzzle easy or difficult? I’m not saying. Place 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. If you use logic to solve the puzzle, you will probably find some surprises around every turn.

Factors of 1738:

It may surprise you that 1738 is divisible by 11. Why is it? Because
1 – 7 + 3 – 8 = -11, a multiple of 11, which makes 1738 divisible by 11.

  • 1738 is a composite number.
  • Prime factorization: 1738 = 2 × 11 × 79.
  • 1738 has no exponents greater than 1 in its prime factorization, so √1738 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 1738 has exactly 8 factors.
  • The factors of 1738 are outlined with their factor pair partners in the graphic below.

More About the Number 1738:

Maybe you will be surprised by some of these number facts about the number 1738.

As mentioned in the table, 1738 = 2 × 11 × 79. That means 1738 is the short leg in the Pythagorean triple calculated from 2(79)(11), 79² – 11², 79² + 11².

1738 is the sum of eleven consecutive numbers:
153 + 154 + 155 + 156 +157 + 158 + 159 + 160 + 161 + 162 + 163 = 1738.

1738 is the sum of four consecutive numbers:
433 + 434 + 435 + 436 = 1738.

And because four is an even number:
436² – 435² + 434² – 433² = 1738.
Surprised?

1738 is also the sum of the 44 consecutive numbers from 18 to 61.

Consequently,
61² – 60² + 59² – 68² + 57² – 56² + 55² – 54² + 53² – 52² + 51² – 50² + 49² – 48² + 47² – 46² + 45² – 44² + 43² – 42² + 41² – 40² + 39² – 38² + 37² – 36² + 35² – 34² + 33² – 32² + 31² – 30² + 29² – 28² + 27² – 26² + 25² – 24² + 23² – 22² + 21² – 20² + 19² – 18² = 1738. I bet you weren’t expecting that!

But the biggest surprise about 1738 is something I learned from OEIS.org: 1738 is in an equation that uses every digit from 1 to 9 exactly one time:
9 Different Digit Equation

make science GIFs like this at MakeaGif