Number

1528 Candy Corn

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Today’s Puzzle:

To solve this Level 3 Candy Corn Halloween puzzle, first, find the factors that will work with the clues in the top and bottom rows. Then work you way down row by row filling in factors as you go.

When you get to the 8 in this puzzle, will the factors be 8 × 1 or 4 × 2? Two of those factors will be eliminated because they already appear in the first column. The other two remain possibilities, but one of those factors cannot appear in any other place in that first column, so that is the one you will want to choose. Have a sweet time solving this puzzle!

Here is a plain version of the same puzzle:

Factors of 1528:

1528 is divisible by two because it is even.

1528 is divisible by four because its last two digits (in the same order) make a number, 28, which is divisible by 4.

Can 1528 be evenly divided by 8? Yes. Here’s a quick way to know: 28 is divisible by 4, but not by 8, AND 5 is odd, so 1528 is divisible by 8, as is every other number ending in 528.

  • 1528 is a composite number.
  • Prime factorization: 1528 = 2 × 2 × 2 × 191, which can be written 1528 = 2³ × 191.
  • 1528 has at least one exponent greater than 1 in its prime factorization so √1528 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1528 = (√4)(√382) = 2√382.
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1528 has exactly 8 factors.
  • The factors of 1528 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1528:

Since 1528 is divisible by 8 but not by 16, it can be written as the sum of 16 consecutive numbers:
88+89+90+91+92+93+94+95+96+97+98+99+100+101+102+103=1528.

Note that 95 + 96 = 191, and 8 × 191 = 1528.
Likewise, 94 + 97 = 1528,
93 + 98 = 1528, and so forth until we get to…
88 + 103 = 1528.

 

 

1527 Not a Haunted House

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Today’s Puzzle:

We see plenty of pumpkins on doorsteps this time of year, so I put a few in this puzzle. The puzzle looks a bit like a house, but certainly not a haunted house. Can you write the numbers from 1 to 10 in both the first column and the top row so that the given number clues are the products of those numbers?

ѼѼѼѼ 🎃🎃🎃🎃 ѽѽѽѽ  

Here’s the same puzzle without the added colorful embellishments:

Factors of 1527:

  • 1527 is a composite number.
  • Prime factorization: 1527 = 3 × 509.
  • 1527 has no exponents greater than 1 in its prime factorization, so √1527 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 1527 has exactly 4 factors.
  • The factors of 1527 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1527:

1527 is the hypotenuse of a Pythagorean triple:
660-1377-1527, which is 3 times (220-459-509).

 

1526 Grave Marker

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Today’s Puzzle:

It’s almost Halloween! This is my favorite kind of grave marker, one that is really just a Find the Factors puzzle in disguise. It’s only a level one, so it isn’t very tricky. I hope you find it a real treat!

Here’s the same puzzle but requiring less ink to print:

 

Neighbors have decorated part of their yard to look like a mini graveyard for Halloween. I think my grave marker would fit right in!

Factors of 1526:

  • 1526 is a composite number.
  • Prime factorization: 1526 = 2 × 7 × 109.
  • 1526 has no exponents greater than 1 in its prime factorization, so √1526 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 1526 has exactly 8 factors.
  • The factors of 1526 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1526:

1526 is the hypotenuse of a Pythagorean triple:
840-1274-1526 which is 14 times (60-91-109)

1525 Challenge Puzzle

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Today’s puzzle:

There is plenty to challenge you in solving this puzzle, but there are also quite a few very helpful clues. Just remember to use logic and have fun with it!

Print the puzzles or type the solution in this excel file: 12 Factors 1511-1525

Factors of 1525:

  • 1525 is a composite number.
  • Prime factorization: 1525 = 5 × 5 × 61, which can be written 1525 = 5² × 61.
  • 1525 has at least one exponent greater than 1 in its prime factorization so √1525 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1525 = (√25)(√61) = 5√61.
  • 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 1525 has exactly 6 factors.
  • The factors of 1525 are outlined with their factor pair partners in the graphic below.

More about the Number 1525:

1525 is the sum of two squares in two different ways:
39² + 2² = 1525
38² + 9² = 1525

1525 is the hypotenuse of a Pythagorean triple in SEVEN different ways!
156-1517-1525, calculated from 2(39)( 2), 39² – 2², 39² + 2²,
275-1500-1525, which is 25 times (11-60-61),
427-1464-1525, which is (7-24-25) times 61,
680-1365-1525, which is 5 times (136-273-305)
684-1363-1525, calculated from 2(38)( 9), 38² – 9², 38² + 9²,
915-1220-1525, which is (3-4-5) times 305, and
1035-1120-1525, which is 5 times (207-224-305).

1525 is also the difference of two squares in two different ways:
763² – 762² = 1525,
155² – 150² = 1525, and
43² – 18² = 1525.

1525 is the 25th heptagonal number because
(5(25²)-3(25))/2 = 1525.
That means you can make a 7-sided figure out of 1525 dots where two of the sides are shared with all the previous heptagons.

1524 Mystery

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Today’s Puzzle:

Is this puzzle easy or difficult? That is the mystery. Will any of the clues trick you into writing factors that won’t work with the rest of the puzzle? If you consistently use logic and not just multiplication and division facts, you’ll solve this puzzle.

Factors of 1524:

  • 1524 is a composite number.
  • Prime factorization: 1524 = 2 × 2 × 3 × 127, which can be written 1524 = 2² × 3 × 127.
  • 1524 has at least one exponent greater than 1 in its prime factorization so √1524 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1524 = (√4)(√381) = 2√381.
  • 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 1524 has exactly 12 factors.
  • The factors of 1524 are outlined with their factor pair partners in the graphic below.

More about the Number 1524:

1524 is the difference of two squares in two different ways:
382² – 380² = 1524,
130² – 124² = 1524.

1523 Mystery Puzzle

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Today’s Puzzle:

My newly turned 10-year-old granddaughter likes making Find the Factors 1 -12 Puzzles with me. Unfortunately, at the beginning of the month, she was in an accident. Her 12-year-old cousin hadn’t ever made a puzzle before but helped me make this one to wish her a speedy recovery. (Thankfully, she is almost fully recovered now.)

Factors of 1523:

  • 1523 is a prime number.
  • Prime factorization: 1523 is prime.
  • 1523 has no exponents greater than 1 in its prime factorization, so √1523 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1523 has exactly 2 factors.
  • The factors of 1523 are outlined with their factor pair partners in the graphic below.

How do we know that 1523 is a prime number? If 1523 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1523. Since 1523 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1523 is a prime number.

Another Fact about the Number 1523:

1523 is the difference of two consecutive squares:
762² – 761² = 1523.

1522 Happy Birthday, Sue, in Spite of All the Chaos!

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Today’s Puzzle:

Today is my sister’s birthday. This year has been a turbulent and chaotic year for many people including her. A hurricane completely damaged her home this summer. Sue, here’s hoping that the coming year will be much brighter for you. Here is a chaotic-looking puzzle for your birthday. If you find all the products after you find all the factors, it will look a lot more orderly. Have a very happy birthday today!

Factors of 1522:

  • 1522 is a composite number.
  • Prime factorization: 1522 = 2 × 761.
  • 1522 has no exponents greater than 1 in its prime factorization, so √1522 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 1522 has exactly 4 factors.
  • The factors of 1522 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1522:

1522 is the sum of two squares:
39² + 1² = 1522.

1522 is the hypotenuse of a Pythagorean triple:
78-1520-1522 calculated from 2(39)(1), 39² – 1² , 39² + 1².

From OEIS.org we learn this curious fact:
The digits of 1522 are 1, 5, 2, and 2.
The squares of each of those digits are 1, 25, 4, and 4.
12544 is a perfect square. It is 112².

 

What Kind of Shape Is 1521 in?

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Today’s Puzzle:

1521 is a perfect square so if you had 1521 tiny squares, you could arrange them into a square perfectly. The following graphic has three sums on the right. The first sum shows that 1521 is the sum of the first 39 odd numbers. The last two sums prove that it is one more than eight times the 19th triangular number.

Since 1521 is one more than eight times the 19th triangular number, it is also the 20th centered octagonal number. Today’s puzzle: How do we arrange 1521 tiny dots into a centered octagon?

I wanted to make a graphic of centered octagonal number 1521. I puzzled over how I could do that for several days. I decided to make it in Desmos. Since octagons are symmetrical, I knew that I could make 1/4 of it in Desmos and duplicate the other quarters in Paint. At first, I spent a lot of time typing ordered pairs in Desmos, until I realized how quickly I could make all the needed ordered pairs in Excel:

I made twenty 2-column ordered pairs in Excel. The top of each column was
(0, 0), (0, 1), (0, 2), etc . After I finished typing all of those in, I typed in the second ordered pair in each column from 2-20. I typed in the numerical value of (√2/2, (n-1) + √2/2) where n was the column number.

Then I used the drag function in Excel to complete columns 3 to 20. I dragged the first two ordered pairs in each column until I got an ordered pair where x = y. I made that last ordered pair be red.

Here is an example of how I build one column of ordered pairs:

I found it easier to do a step for every column of ordered pairs before moving onto the next step. Since I wanted these points AND their inverses, I copied and pasted the non-red numbers from the x column into the y column and vice-versa.

Then I used the sort feature in Excel to get all the ordered pairs in ascending order of the x’s.

Lastly, I copied and pasted each column of ordered pairs in Desmos, and my quarter centered octagon quickly grew bigger and bigger with each paste. I found it quite enjoyable to do!

My octagon has lines and numbers on it and it wasn’t symmetrical so I needed to clean up the picture. I clicked on the wrench in Desmos, clicked on “Zoom Square” and removed all the checkmarks I saw. Then I clicked on the gear found above the ordered pairs in Desmos, clicked on the colored circles for each set of ordered pairs, and then clicked on the circle next to the word “lines,” and chose the solid line.

After I put the quarter octagon into Paint, I rotated copies of it every which way, added some words and brown lines until I got this beauty:

It was so much fun to make, I encourage you to give it a try! I bet you can clearly see that 1521 is one more than eight times the 19th triangular number!

Factors of 1521:

  • 1521 is a composite number.
  • Prime factorization: 1521 = 3 × 3 × 13 × 13, which can be written 1521 = 3²× 13².
  • 1521 has at least one exponent greater than 1 in its prime factorization so √1521 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1521 = (√9)(√169) = (3)(13) = 39.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1521 has exactly 9 factors.
  • The factors of 1521 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1521:

1521 is the sum of two squares:
36² + 15² = 1521.

1521 is the hypotenuse of two Pythagorean triples:
585-1404-1521, which is (5-12-13) times 117, and
1071-1080-1521, calculated from 36² – 15², 2(36)(15), 36² + 15².

From OEIS.org we learn that 1521 is sum of 4 distinct cubes in 3 different ways:
11³ +  5³ +  4³ +  1³ = 1521,
10³ +  8³ +  2³ +  1³ = 1521, and
9³ +  8³ +  6³ +  4³ = 1521.

We also learn that 1520 and 1521 are a Ruth-Aaron pair because they are consecutive numbers and the sum of their factors are equal to each other:
1520 = 2⁴·5·19 and 1521 = 3²·13², while
2+2+2+2+5+19 = 32=3+3+13+13.

What Kind of Shape is 1520 in?

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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.

From OEIS.org we learn that 1520 and 1521 are a Ruth-Aaron pair because they are consecutive numbers and the sum of their factors are equal to each other:
1520 = 2⁴·5·19 and 1521 = 3²·13², while
2+2+2+2+5+19 = 32=3+3+13+13.

 

 

 

What Kind of Shape is 1519 in?

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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.