Celebrating 1500 with a Horse Race and Much More!

Pick Your Pony!

Writing 1500 posts is quite a milestone. I’ll begin the celebration with an exciting horse race! Let me explain:

Each prime number has exactly 2 factors. Every composite number between 1401 and 1500 has somewhere between 4 and 36 factors. Which quantity of factors do you think will appear most often for these numbers? Pick that amount as your pony and see how far it gets in this horse race!
1401 to 1500 Horse Race

make science GIFs like this at MakeaGif

Did the race results surprise you? They surprised me!

Prime Factorization for numbers from 1401 to 1500:

Here’s a chart showing the prime factorization of all those numbers and the amount of factors each number has. Numbers in pink have exponents in their prime factorization so their square roots can be simplified:

Today’s Puzzles:

Let’s continue the celebration with a puzzle: 1500 has 12 different factor pairs. One of those pairs adds up to 85 and one of them subtracts to give 85. Can you find those factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1500 in the graphic after the puzzle, but the second puzzle is really just the first puzzle in disguise. So try solving that easier puzzle first.

Factors of 1500:

  • 1500 is a composite number.
  • Prime factorization: 1500 = 2 × 2 × 3 × 5 × 5 × 5, which can be written 1500 = 2² × 3 × 5³
  • 1500 has at least one exponent greater than 1 in its prime factorization so √1500 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1500 = (√100)(√15) = 10√15
  • The exponents in the prime factorization are 2, 1, and 3. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(3 + 1) = 3 × 2 × 4 = 24. Therefore 1500 has exactly 24 factors.
  • The factors of 1500 are outlined with their factor pair partners in the graphic below.

A Forest of Factor Trees:

Take a few minutes to hike in this forest featuring a few of the MANY possible factor trees for 1500. Celebrate each tree’s uniqueness!

Other Facts to Celebrate about 1500:

Oeis.org tells us that  (5+1) × (5+5) × (5+0) × (5+0) = 1500.

1500 is the hypotenuse of THREE Pythagorean triples:
420-1440-1500, which is (7-24-25) times 60,
528-1404-1500, which is 12 times (44-117-225),
900-1200-1500, which is (3-4-5) times 300.

1499 Challenge Puzzle

Today’s Puzzle:

Use the 19 clues, logic, and the multiplication facts from a 10 × 10 multiplication table to find the unique solution of this Find the Factors Challenge puzzle. Good luck!

Factors of 1499:

  • 1499 is a prime number.
  • Prime factorization: 1499 is prime.
  • 1499 has no exponents greater than 1 in its prime factorization, so √1499 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1499 has exactly 2 factors.
  • The factors of 1499 are outlined with their factor pair partners in the graphic below.

How do we know that 1499 is a prime number? If 1499 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1499. Since 1499 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1499 is a prime number.

More Facts about the Number 1499:

1499 is the difference of two consecutive squares:
750² – 749² = 1499

oeis.org reminds us that 149, 199, and 499 are also prime numbers, so taking away one digit from 1499 always leaves a prime number.

 

 

1498 Another Mystery

Today’s Puzzle:

The number 48 appears four times in a 12 × 12 multiplication table, and all four 48’s appear in this puzzle! Where will you put its factors, 6, 8, 4, 12? Use logic to figure out where all the numbers from 1 to 12 need to go to make this puzzle turn into a multiplication table.

Factors of 1498:

  • 1498 is a composite number.
  • Prime factorization: 1498 = 2 × 7 × 107.
  • 1498 has no exponents greater than 1 in its prime factorization, so √1498 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 1498 has exactly 8 factors.
  • The factors of 1498 are outlined with their factor pair partners in the graphic below.

Pythagorean Triples with 1498:

1498 is 2(107)(7) so we can calculate a cool Pythagorean triple from
2(107)(7), 107² – 7², 107² + 7² to get 1498-11400-11498.

Also from 2(749)(1), 749² – 1², 749² + 1², we get 1498-561000-561002.

 

1497 Mystery

Today’s Puzzle:

Knowing where to place some of the numbers from 1 to 12 in this puzzle shouldn’t be too difficult, but placing ALL of the numbers will be like solving a mystery.

Factors of 1497:

  • 1497 is a composite number.
  • Prime factorization: 1497 = 3 × 499
  • 1497 has no exponents greater than 1 in its prime factorization, so √1497 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 1497 has exactly 4 factors.
  • The factors of 1497 are outlined with their factor pair partners in the graphic below.

More about the Number 1497:

1497 is the difference of two squares in two different ways:
749² – 748² = 1497
251² – 248² = 1497

1496 is a Square Pyramidal Number

Visualize 1496 Blocks:

I hoped to make a graphic illustrating that 1496 is the 16th square pyramidal number. I am thrilled that I succeeded!

Factors of 1496:

Knowing the multiplication table and some divisibility tricks helped me find some of 1496’s factors:
96 is 8 × 12, and 4 is even, so 1496 is divisible by 8.
1 – 4 + 9 – 6 = 0, so 1496 is divisible by 11.

  • 1496 is a composite number.
  • Prime factorization: 1496 = 2 × 2 × 2 × 11 × 17, which can be written 1496 = 2³ × 11 × 17
  • 1496 has at least one exponent greater than 1 in its prime factorization so √1496 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1496 = (√4)(√374) = 2√374
  • 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 1496 has exactly 16 factors.
  • The factors of 1496 are outlined with their factor pair partners in the graphic below.

Another fact about the number 1496:

1496 is the hypotenuse of a Pythagorean triple:
704-1320-1496, which is (8-15-17) times 88.

1495 and Level 6

Today’s Puzzle:

Hint: The only way 12 can be put in the first column of this puzzle is to let one of the 60’s use it. We don’t have to know which 60 that is, to know that a 5 will go above that 60 in the top row. Knowing that, where does 5 have to go in the first column? That’s the logic needed to get started on this puzzle:

Factors of 1495:

  • 1495 is a composite number.
  • Prime factorization: 1495 = 5 × 13 × 23.
  • 1495 has no exponents greater than 1 in its prime factorization, so √1495 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 1495 has exactly 8 factors.
  • The factors of 1495 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1495:

1495 is the hypotenuse of FOUR Pythagorean triples:
368-1449-1495, which is 23 times (16-63-65)
575-1380-1495, which is (5-12-13) times 115,
759-1288-1495, which is 23 times (33-56-65), and
897-1196-1495, which is (3-4-5) times 299.

 

 

1494 and Level 5

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 those factors. Be sure to use logic to find the factors! Guessing and checking will only frustrate you.

Factors of 1494:

  • 1494 is a composite number.
  • Prime factorization: 1494 = 2 × 3 × 3 × 83, which can be written 1494 = 2 × 3² × 83
  • 1494 has at least one exponent greater than 1 in its prime factorization so √1494 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1494 = (√9)(√166) = 3√166
  • 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 1494 has exactly 12 factors.
  • The factors of 1494 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1494:

1494 is not the sum of or the difference of two squares, but it is still a part of three Pythagorean triples because of these three ways it can be factored:
1494 = 2(747)(1),
1494 = 2(249)(3), and
1494 = 2(83)(9).
And because for whole numbers where a > b, 2(a)(b), a² – b², a² + b² will be a Pythagorean triple.

1493 and Level 4

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given in the puzzle:

Factors of 1493:

  • 1493 is a prime number.
  • Prime factorization: 1493 is prime.
  • 1493 has no exponents greater than 1 in its prime factorization, so √1493 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1493 has exactly 2 factors.
  • The factors of 1493 are outlined with their factor pair partners in the graphic below.

How do we know that 1493 is a prime number? If 1493 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1493. Since 1493 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1493 is a prime number.

Other Facts about the number 1493:

1493 is the last prime number in the fourth prime quintuplet,
(1481, 1483, 1487, 1489, 1493), which is the smallest prime quintuplet that is not also part of a prime sextuplet.
In prime quintuplets, the first three numbers, the middle three numbers, and the last three numbers each form a prime triplet. Thus,1493 is the last prime number in the third prime triplet formed from the numbers in the fourth prime quintuplet.

1493 is the sum of two squares:
38² + 7² = 1493

1493 is the hypotenuse of a Pythagorean triple:
532-1395-1493, calculated from 2(38)(7), 38² – 7², 38² + 7²

Here’s another way we know that 1493 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 7² = 1493 with 38 and 7 having no common prime factors, 1493 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1493. Since 1493 is not divisible by 5, 13, 17, 29, or 37, we know that 1493 is a prime number.

1492 The Niña, the Pinta, and the Santa Maria

Today’s Puzzle:

I debated with myself about whether or not I should mention the famous poem about 1492 in this post. Then I looked at the puzzle I had already created for today and noticed it would not be too much of a stretch to say the twelve clues look a little bit like three ships.  So whether or not I should, I decided to go ahead and mention the Niña, the Pinta, and the larger Santa Maria in the title. The ships themselves are innocent of any barbaric acts and might just be the three most famous ships in world history.

Start at the top of this level 3 puzzle and work your way down cell by cell using logic until you have written all the factors from 1 to 12 in both the first column and the top row.

Here’s the same puzzle without color:

Factors of 1492:

  • 1492 is a composite number.
  • Prime factorization: 1492 = 2 × 2 × 373, which can be written 1492 = 2² × 373
  • 1492 has at least one exponent greater than 1 in its prime factorization so √1492 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1492 = (√4)(√373) = 2√373
  • 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 1492 has exactly 6 factors.
  • The factors of 1492 are outlined with their factor pair partners in the graphic below.

Other Facts About the Number 1492:

1492 is the sum of two squares:
36² + 14² = 1492

1492 is the hypotenuse of a Pythagorean triple:
1008-1100-1492, which is 4 times (252-275-373)
and can also be calculated from 2(36)(14), 36² – 14², 36² + 14².

1491 How Many Square Units Do You See in This Partially Completed Puzzle?

Today’s Puzzle:

My grandchildren had put this puzzle together many times on their own but wanted me to help them this time. The picture on the box was very helpful and let me know there were four unicorns, but I could only find puzzle pieces for three horns instead of four. I found the missing horn under some furniture in the living room, but I still wondered if any other puzzle pieces had been misplaced. I used multiplication, addition, and subtraction to figure out how many puzzle pieces still needed to be placed on this partially completed puzzle. Can you also figure out the number of puzzle pieces that still need to be placed?

I counted the number of puzzle pieces in the box and determined that two pieces were missing. Knowing that two puzzle pieces are missing, can you tell me how many more puzzle pieces are in the box waiting to be placed on the puzzle?

Each puzzle piece is approximately equal to one square unit. If you can determine the area of the entire puzzle and the area of the missing pieces, then you can determine the area of the incomplete puzzle pictured above. Go ahead, give it a try!

Factors of 1491:

  • 1491 is a composite number.
  • Prime factorization: 1491 = 3 × 7 × 71.
  • 1491 has no exponents greater than 1 in its prime factorization, so √1491 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 1491 has exactly 8 factors.
  • The factors of 1491 are outlined with their factor pair partners in the graphic below.

Another Fact about the number 1491:

1491 is the 21st nonagonal number because
21(7·21 – 5)/2 = 1491.