# 1546 Celebrating My 7th Blogiversary

### Today’s Puzzle:

Seven years ago today at 2:11 in the morning, I published my first blog post. It featured a Find the Factors 1 – 10 puzzle with only perfect square clues. In honor of that first post, I publish this puzzle, also with only perfect square clues. This one looks like an exclamation point because I have enjoyed creating puzzles and writing these posts for you so much these seven years! How do you solve the puzzle? Use logic to write the numbers from 1 to 10 in both the first column and the top row so that the puzzle functions like a multiplication table.

### Factors of 1546:

• 1546 is a composite number.
• Prime factorization: 1546 = 2 × 773.
• 1546 has no exponents greater than 1 in its prime factorization, so √1546 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 1546 has exactly 4 factors.
• The factors of 1546 are outlined with their factor pair partners in the graphic below. ### More about the Number 1546:

1546 is the sum of two squares:
39² + 5² = 1546.

1546 is the hypotenuse of a Pythagorean triple:
390-1496-1546, calculated from 2(39)(5), 39² – 5², 39² + 5².
It is also 2 times (195-748-773)

# 1542 Fourth Letter of the Message

### Today’s Puzzle:

This level one puzzle is the easiest of all the puzzles in my message to you. It might also be the letter that helps you know exactly what the message says. ### Factors of 1542:

• 1542 is a composite number.
• Prime factorization: 1542 = 2 × 3 × 257.
• 1542 has no exponents greater than 1 in its prime factorization, so √1542 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 1542 has exactly 8 factors.
• The factors of 1542 are outlined with their factor pair partners in the graphic below. ### Another Fact about the Number 1542

1542 is the hypotenuse of a Pythagorean triple:
192-1530-1542, which is 6 times (32-255-257).

# 1526 Grave Marker

### 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)

# 1511 and Level 1

### Today’s Puzzle:

Do you know what number is the common factor of all the clues in this puzzle? If you do, then you are well on your way to solving this level one puzzle! ### Factors of 1511:

• 1511 is a prime number.
• Prime factorization: 1511 is prime.
• 1511 has no exponents greater than 1 in its prime factorization, so √1511 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1511 has exactly 2 factors.
• The factors of 1511 are outlined with their factor pair partners in the graphic below. How do we know that 1511 is a prime number? If 1511 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1511. Since 1511 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1511 is a prime number.

### One More Fact about the Number 1511:

1511 is equidistant from the two closest primes:
The previous prime was 1511 – 12 = 1499.
The next prime will be 1511 + 12 = 1523.

# 1502 and Level 1

### Today’s Puzzle:

This level 1 puzzle has products in one of the rows and in one of the columns. Can you use those products to figure out where the factors 1 to 10 belong in this multiplication table puzzle? ### Factors of 1502:

• 1502 is a composite number.
• Prime factorization: 1502 = 2 × 751
• 1502 has no exponents greater than 1 in its prime factorization, so √1502 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 1502 has exactly 4 factors.
• The factors of 1502 are outlined with their factor pair partners in the graphic below. ### Pythagorean Triple with 1502:

1502 is not the sum or the difference of two squares, but 1502 = 2(751)(1), so it is part of a Pythagorean triple:
1502-564000-564002, calculated from 2(751)(1), 751² – 1², 751² – 1².

# 1489 and Level 1

### Today’s Puzzle:

You can use the 20 clues in this puzzle to find the order that the factors from 1 to 12 must take. After you find those factors you can multiply each pair of factors together to complete the entire multiplication table. ### Factors of 1489:

• 1489 is a prime number.
• Prime factorization: 1489 is prime.
• 1489 has no exponents greater than 1 in its prime factorization, so √1489 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1489 has exactly 2 factors.
• The factors of 1489 are outlined with their factor pair partners in the graphic below. How do we know that 1489 is a prime number? If 1489 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1489. Since 1489 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1489 is a prime number.

### Other Facts about the Number 1489:

1489 is the sum of two squares:
33² + 20² = 1489.

1489 is the hypotenuse of a Pythagorean triple:
689-1320-1489 calculated from 33² – 20², 2(33)(20), 33² + 20².

Here’s another way we know that 1489 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 33² + 20² = 1489 with 33 and 20 having no common prime factors, 1489 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1489. Since 1489 is not divisible by 5, 13, 17, 29, or 37, we know that 1489 is a prime number.

1489 is the last prime number in the fifth prime decade: 1489 is a centered triangular number because (3(31²) + 3(31) + 2)/2 = 1489. # 1478 and Level 1

### Today’s Puzzle:

If you can divide a few even numbers in half, then you can find all the factors needed to solve this puzzle. Then you can complete the multiplication table that the puzzle makes. Go for it! ### Factors of 1478:

• 1478 is a composite number.
• Prime factorization: 1478 = 2 × 739
• 1478 has no exponents greater than 1 in its prime factorization, so √1478 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 1478 has exactly 4 factors.
• The factors of 1478 are outlined with their factor pair partners in the graphic below. ### A little more about the number 1478:

1478 is not the sum or the difference of two squares, but it is the short leg in a Pythagorean triple:
(1478-546120-546122) calculated from 2(739)(1), 739² – 1², 739² + 1².

# 1468 and Level 1

### Today’s Puzzle

Can you find the factors from 1 to 12 that will work with the given clues to turn this puzzle into a multiplication table? ### Factors of 1468

• 1468 is a composite number.
• Prime factorization: 1468 = 2 × 2 × 367, which can be written 1468 = 2² × 367
• 1468 has at least one exponent greater than 1 in its prime factorization so √1468 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1468 = (√4)(√367) = 2√367
• 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 1468 has exactly 6 factors.
• The factors of 1468 are outlined with their factor pair partners in the graphic below. ### Another Interesting Fact about the number 1468:

OEIS.org informs us that the smallest 20-digit 6th power is 1468⁶.

# 1455 and Level 1

### Today’s Puzzle:

All of the clues in today’s puzzle are divisible by the same number. Can you figure out what that number is? If you can, then you can solve this puzzle. ### Factors of 1455:

That was puzzle number 1455. In case you would like to know a little bit about that number, here are a few facts:

• 1455 is a composite number.
• Prime factorization: 1455 = 3 × 5 × 97
• 1455 has no exponents greater than 1 in its prime factorization, so √1455 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 1455 has exactly 8 factors.
• The factors of 1455 are outlined with their factor pair partners in the graphic below. ### Facts about the number 1455:

1455 is the difference of two squares in four different ways:
728² – 727² = 1455
244² – 241² = 1455
148² – 143² = 1455
56² – 41² = 1455

1455 is also the hypotenuse of four Pythagorean triples:
132-1449-1455 which is 3 times (44-483-485)
279-1428-1455 which is 3 times (93-476-485)
873-1164-1455 which is (3-4-5) times 291
975-1080-1455 which is 15 times (65-72-97)

# 1443 A Simple Gift for You

This level one puzzle is my simple gift to you. Yes, you can solve it, and you don’t even have to wait until December 25th to discover all the factors and products to be found inside! That was puzzle number 1443. Here are some facts about that number.

• 1443 is a composite number.
• Prime factorization: 1443 = 3 × 13 × 37
• 1443 has no exponents greater than 1 in its prime factorization, so √1443 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 1443 has exactly 8 factors.
• The factors of 1443 are outlined with their factor pair partners in the graphic below. (38-1)(38+1) = 1443 so 1443 is one number away from 38² = 1444.

Actually, 1443 is the difference of two squares in four different ways:
38² – 1² = 1443
62²- 49² = 1443
242² – 239² = 1443
722² – 721² = 1443

1443 is the hypotenuse of FOUR Pythagorean triples:
93-1440-1443 which is 3 times (31-480-481)
468-1365-1443 which is (12-35-37) times 39
555-1332-1443 which is (5-12-13) times 111
957-1080-1443 which is 3 times(319-360-481)