What Kind of Shape is 1520 in?

Today’s Puzzle:

Sure, it’s a rectangle with whole-number sides in 10 different ways, but what kind of REGULAR polygonal shape can 1520 be made into? I will tell you that the measurement of each of its sides is 32.

And thus, it is the 32nd shape of its kind. By the way, I really like how all the 32nd figurate numbers relate to each other:

We see in the chart that 1520 dots can be arranged into a pentagon. Just how do we do that? Here’s how:

Do you see from the graphic that 1520 is 32 more than three times the 31st triangular number?

1520 is also related to triangular numbers in another way: Today I learned that all pentagonal numbers are 1/3 of a triangular number.  Indeed, 1520 is 1/3 of the 95th triangular number:
(1/3) of (95)(96)/2 = 1520.

Pretty cool, I think!

Factors of 1520:

  • 1520 is a composite number.
  • Prime factorization: 1520 = 2 × 2 × 2 × 2 × 5 × 19, which can be written 1520 = 2⁴ × 5 × 19
  • 1520 has at least one exponent greater than 1 in its prime factorization so √1520 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1520 = (√16)(√95) = 4√95. The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1520 has exactly 20 factors.
  • The factors of 1520 are outlined with their factor pair partners in the graphic below.

More about the Number 1520:

Recently someone on twitter asked:

If you look at the whole thread, you will see how a few people explained this important concept using arrays. Here is my attempt to explain the difference of two squares using 1520 and arrays:

As I mentioned before, 1520 has 10 rectangles with whole-number sides. The one with the smallest perimeter is 38 × 40, and it is the easiest to use to demonstrate how 1520 is the difference of two squares:

1520 Difference of Two Squares

make science GIFs like this at MakeaGif

 

I made that gif be as slow as I could without duplicating any of the frames, but it still goes pretty fast.

1520 is, in fact, the difference of two squares in six different ways:
39² – 1² = 1520,
48² – 28² = 1520,
81² – 71² = 1520,
99² – 91² = 1520,
192² – 188² = 1520, and
381² – 379² = 1520.

1520 is also the hypotenuse of a Pythagorean triple:
912-1216-1520, which is (3-4-5) times 304.

 

 

 

What Kind of Shape is 1519 in?

Today’s Puzzle:

1519, 1520, and 1521 are all figurate numbers. What kind of shape can you arrange 1519 tiny dots?

1519 is the 23rd centered hexagonal number because 23³ – 22³ = 1519.

It is also the 23rd centered hexagonal number because it is one more than six times the 23rd triangular number. Do you see the 23rd triangular number six times in the graphic above?

Factors of 1519:

  • 1519 is a composite number.
  • Prime factorization: 1519 = 7 × 7 × 31, which can be written 1519 = 7² × 31
  • 1519 has at least one exponent greater than 1 in its prime factorization so √1519 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1519 = (√49)(√31) = 7√31
  • 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 1519 has exactly 6 factors.
  • The factors of 1519 are outlined with their factor pair partners in the graphic below.

1518 and Level 6

Today’s Puzzle:

Level 6 puzzles are designed to be a little tricky. Just make sure you use logic to figure out the factors every time, and you will get it done!

Factors of 1518:

  • 1518 is a composite number.
  • Prime factorization: 1518 = 2 × 3 × 11 × 23
  • 1518 has no exponents greater than 1 in its prime factorization, so √1518 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 1518 has exactly 16 factors.
  • The factors of 1518 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1518:

1518 has a palindromic prime factorization. (The digits are the same frontward or backward.)
1518 = 2 · 3 · 11 · 23

1517 and Level 5

Today’s Puzzle:

Which common factor of 72 and 36 is needed to solve this puzzle? Is it 6, 9, or 12? There is an easier place to begin this level 5 puzzle. Don’t guess and check. Use logic to know which factors you should use.  You can figure it out!

Factors of 1517:

  • 1517 is a composite number.
  • Prime factorization: 1517 = 37 × 41
  • 1517 has no exponents greater than 1 in its prime factorization, so √1517 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 1517 has exactly 4 factors.
  • The factors of 1517 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1517:

1517 is the difference of two squares in two different ways:
759² – 758² = 1517,
39² – 2² = 1517.

1517 is also the sum of two squares in two different ways:
34² + 19² = 1517,
29² + 26² = 1517.

1517 is the hypotenuse of FOUR Pythagorean triples:
165-1508-1517, calculated from 29² – 26², 2(29)(26), 29² + 26²,
333-1480-1517, which is 37 times (9-40-41),
492-1435-1517, which is (12-35-37) times 41,
795-1292-1517, calculated from 34² – 19², 2(34)(19), 34² + 19².

1516 and Level 4

Today’s Puzzle:

Using logic, 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 1516:

  • 1516 is a composite number.
  • Prime factorization: 1516 = 2 × 2 × 379, which can be written 1516 = 2² × 379
  • 1516 has at least one exponent greater than 1 in its prime factorization so √1516 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1516 = (√4)(√379) = 2√379
  • 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 1516 has exactly 6 factors.
  • The factors of 1516 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1516:

Only one of its factor pairs add up to an even number, so 1516 is the difference of two squares in only one way:
380² – 378² = 1516.

1515 and Level 3

Today’s Puzzle:

Start with the greatest common factor of 30 and 48, write the factors in the appropriate boxes, then work your way down this level 3 puzzle row by row using logic until you have found all the factors. You can do this!

Factors of 1515:

  • 1515 is a composite number.
  • Prime factorization: 1515 = 3 × 5 × 101.
  • 1515 has no exponents greater than 1 in its prime factorization, so √1515 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 1515 has exactly 8 factors.
  • The factors of 1515 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1515:

1515 is the hypotenuse of FOUR Pythagorean triples:
300-1485-1515, which is 15 times (20-99-101),
651-1368-1515, which is 3 times (217-456-505),
909-1212-1515, which is (3-4-5) times 303,
1008-1131-1515, which is 3 times (336-377-505).

1514 and Level 2

Today’s Puzzle:

There is only one way to write the factors from 1 to 12 in both the first column and the top row so that this puzzle will behave like a multiplication table. The given clues will be the products of the factors you write. Can you find the way?

Factors of 1514:

  • 1514 is a composite number.
  • Prime factorization: 1514 = 2 × 757
  • 1514 has no exponents greater than 1 in its prime factorization, so √1514 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 1514 has exactly 4 factors.
  • The factors of 1514 are outlined with their factor pair partners in the graphic below.

More Facts about the number 1514:

1514 is the sum of two squares:
35² + 17² = 1514

1514 is the hypotenuse of a Pythagorean triple:
936-1190-1514 calculated from 35² – 17², 2(35)(17), 35² + 17²

1513 is the Sum of Squares

Today’s Puzzle:

How can you arrange 1513 dots into a perfect square when √1513 is irrational?

The answer is you arrange the dots into a centered square like this:

You can arrange them like that because 1513 is the sum of consecutive squares.

Factors of 1513:

  • 1513 is a composite number.
  • Prime factorization: 1513 = 17 × 89
  • 1513 has no exponents greater than 1 in its prime factorization, so √1513 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 1513 has exactly 4 factors.
  • The factors of 1513 are outlined with their factor pair partners in the graphic below.

More about the Number 1513:

1513 is the sum of two squares in two different ways:
28² + 27²  = 1513, and
37² + 12²  = 1513.

1513 is the hypotenuse of FOUR Pythagorean triples:
55-1512-1513,  calculated from 28² – 27², 2(28)( 27), 28² + 27²
663-1360-1513, which is 17 times (39-80-89)
712-1335-1513, which is (8-15-17) times 89
888-1225-1513, calculated from 2(37)(12), 37² – 12², 37² + 12²

Could 1513 be a prime number?

Since its last two digits divided by 4 leave a remainder of 1, and 28² + 27² = 1513 with   28 and 27 having no common prime factors, 1513 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1513. Is 1513 divisible by 5, 13, 17, 29, or 37? Yes, it is divisible by 17, so 1513 is NOT a prime number.

37² + 12²  = 1513 and 37 and 12 have no common prime factors, so we could have arrived at the same result using those numbers.

Note: Numbers that are the sum of two squares in two or more ways are never prime.

 

1510 Challenge Puzzle

Today’s Puzzle:

Challenge puzzles are like four multiplication tables connected to each other. Use logic to place the factors 1 to 10 in each boldly outlined column or row so that the given clues are the products of the factors you write. I hope you enjoy solving this puzzle as much as I enjoyed making it for you!

Here’s an excel file with this week’s puzzles: 10 Factors 1502-1510

Factors of 1510:

  • 1510 is a composite number.
  • Prime factorization: 1510 = 2 × 5 × 151.
  • 1510 has no exponents greater than 1 in its prime factorization, so √1510 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 1510 has exactly 8 factors.
  • The factors of 1510 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1510:

1510 is the hypotenuse of a Pythagorean triple:
906-1208-1510, which is (3-4-5) times 302.

 

1509 A Mystery Level Puzzle on Training Wheels

Today’s Puzzle:

I have described level 3 puzzles as level 4 puzzles on training wheels. Today’s puzzle is definitely not a level 3 puzzle, but it is on training wheels. A logical way to solve this puzzle is to start with the clue in the top row and work your way down the puzzle row by row writing the factors as you go. Giving you the logical order to use the clues should help some, but the logic needed to find the factors will still be a mystery. Don’t guess and check. Please, use logic! Think about how each clue relates to the other clues in the puzzle.

Factors of 1509:

  • 1509 is a composite number.
  • Prime factorization: 1509 = 3 × 503
  • 1509 has no exponents greater than 1 in its prime factorization, so √1509 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 1509 has exactly 4 factors.
  • The factors of 1509 are outlined with their factor pair partners in the graphic below.

More about the Number 1509:

1509 is the difference of two squares in two different ways:
755² – 754² = 1509
253² – 250² = 1509