Puzzle

1531 Spider’s Web or Not?

/ ivasallay / Leave a comment

Today’s Puzzle:

Even though1531 is a prime number, what geometric shape can you arrange 1531 tiny squares into?

As you look at the graphic below, ask yourself a couple of questions. What do you notice? What do you wonder?

You might think that you are looking at a spider’s web with the numbers 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, and 1531 trapped inside.

All of those numbers are centered decagonal numbers. My puzzle for you today is: If it were a spider’s web, and a spider ate the last digit of each of those centered decagonal numbers, what kind of figurate number would be left behind in every case?

Factors of 1531:

  • 1531 is a prime number.
  • Prime factorization: 1531 is prime.
  • 1531 has no exponents greater than 1 in its prime factorization, so √1531 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1531 has exactly 2 factors.
  • The factors of 1531 are outlined with their factor pair partners in the graphic below.

How do we know that 1531 is a prime number? If 1531 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1531. Since 1531 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1531 is a prime number.

Another Fact about the Number 1531:

OEIS.org informs us that 1531⁴ = 5494153169521.

1530 Jack-o’-lantern

/ ivasallay / Leave a comment

Today’s Puzzle:

Here’s a Jack-O’-Lantern Puzzle for you to enjoy. It’s a Level 5 puzzle so it might be more of a trick than a treat. Remember to use logic every step of the way instead of guessing and checking.

Here’s the same puzzle without any added color:

Factors of 1530:

15 is half of 30, so 1530 is divisible by 6 just like all these numbers are divisible by 6: 12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, and so forth.

  • 1530 is a composite number.
  • Prime factorization: 1530 = 2 × 3 × 3 × 5 × 17, which can be written 1530 = 2 × 3² × 5 × 17.
  • 1530 has at least one exponent greater than 1 in its prime factorization so √1530 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1530 = (√9)(√170) = 3√170.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1530 has exactly 24 factors.
  • The factors of 1530 are outlined with their factor pair partners in the graphic below.

More about the Number 1530:

51 × 30 = 1530. Did you notice that the same digits appear on both sides of the equal sign and only +, -, ×, ÷, (), or exponents were used to make a true statement? 1530 is only the 25th number that can make that claim, so we call it the 25th Friedman number.

There are MANY possible factor trees for 1530, but let’s celebrate that it is also a Friedman number with this one:

1530 is the hypotenuse of FOUR Pythagorean triples:
234-1512-1530, which is 18 times (13-84-85),
648-1386-1530, which is 18 times (36-77-85),
720-1350-1530, which is (8-15-17) times 90, and
918-1224-1530, which is (3-4-5) times 306.

 

 

1529 Pointy Hat

/ ivasallay / Leave a comment

Today’s Puzzle:

A pointy hat is part of many different Halloween costumes: Wizards, Witches, Medieval Princesses, and Clowns come to my mind. Today’s puzzle looks like a pointy hat. Use logic to work your magic in solving it! As always, there is only one solution.

Factors of 1529:

Look at this math fact using the digits of 1529:
1 – 5 + 2 – 9 = -11, a number divisible by 11, so 1529 is divisible by 11.

  • 1529 is a composite number.
  • Prime factorization: 1529 = 11 × 139.
  • 1529 has no exponents greater than 1 in its prime factorization, so √1529 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 1529 has exactly 4 factors.
  • The factors of 1529 are outlined with their factor pair partners in the graphic below.

More about the Number 1529:

1529 is the difference of two squares in two different ways:
765² – 764²  = 1529, and
75² – 64²  = 1529.

 

1528 Candy Corn

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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!

/ ivasallay / Leave a comment

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.