1553 Ornamental Corn

Today’s Puzzle:

Ornamental corn is a popular decoration at Thanksgiving. Today’s puzzle looks a little bit like ornamental corn, and there’s at least a kernel of truth to that statement! Solve the puzzle, and I will think YOU are a-maize-ing!

Here’s the same puzzle if you want to print it in black and white:

Factors of 1553:

  • 1553 is a prime number.
  • Prime factorization: 1553 is prime.
  • 1553 has no exponents greater than 1 in its prime factorization, so √1553 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1553 has exactly 2 factors.
  • The factors of 1553 are outlined with their factor pair partners in the graphic below.

How do we know that 1553 is a prime number? If 1553 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1553. Since 1553 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1553 is a prime number.

More About the Number 1553:

1553 is the sum of the squares of two numbers that are reverses of each other:
32² + 23² = 1553

1553 is the hypotenuse of a primitive Pythagorean triple:
495-1472-1553, calculated from 32² – 23², 2(32)(23), 32² + 23².

Here’s another way we know that 1553 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 23² = 1553 with 32 and 23 having no common prime factors, 1553 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1553. Since 1553 is not divisible by 5, 13, 17, 29, or 37, we know that 1553 is a prime number.

1549 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that the given clues are the products of the numbers you write.

Factors of 1549:

  • 1549 is a prime number.
  • Prime factorization: 1549 is prime.
  • 1549 has no exponents greater than 1 in its prime factorization, so √1549 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1549 has exactly 2 factors.
  • The factors of 1549 are outlined with their factor pair partners in the graphic below.

How do we know that 1549 is a prime number? If 1549 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1549. Since 1549 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1549 is a prime number.

More about the number 1549:

1549 is the sum of two squares:
35² + 18² = 1549

1549 is the hypotenuse of a Pythagorean triple:
901-1260-1549, calculated from 35² – 18², 2(35)(18), 35² + 18².

Here’s another way we know that 1549 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 18² = 1549 with 35 and 18 having no common prime factors, 1549 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1549. Since 1549 is not divisible by 5, 13, 17, 29, or 37, we know that 1549 is a prime number.

1543 Another Letter

Today’s Puzzle:

My goal was to make this next letter of my message using as few clues in the puzzle as possible. You now have enough clues to solve this puzzle and know my message to you.

Factors of 1543:

  • 1543 is a prime number.
  • Prime factorization: 1543 is prime.
  • 1543 has no exponents greater than 1 in its prime factorization, so √1543 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1543 has exactly 2 factors.
  • The factors of 1543 are outlined with their factor pair partners in the graphic below.

How do we know that 1543 is a prime number? If 1543 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1543. Since 1543 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1543 is a prime number.

One More Fact about the Number 1543:

1543 is the sum of consecutive numbers as well as the difference of those same consecutive numbers but squared:
771 + 772 = 1543;
772² – 771² = 1543.
(1543 has that property because it’s an odd number greater than 1.)

1531 Spider’s Web or Not?

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 1531 = 5494153169521.

1523 Mystery Puzzle

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.

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.

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.

 

 

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.

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.

1487 A Challenging Puzzle

Today’s Puzzle:

This puzzle is like four out-of-order 1 – 10 multiplication tables that work together. The clues in the puzzle need to be the products of the numbers you write. It won’t be easy, but use logic to solve it. Good luck!

Print the puzzles or type the solution in this excel file: 10 Factors 1478-1487

Factors of 1487:

  • 1487 is a prime number.
  • Prime factorization: 1487 is prime.
  • 1487 has no exponents greater than 1 in its prime factorization, so √1487 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1487 has exactly 2 factors.
  • The factors of 1487 are outlined with their factor pair partners in the graphic below.

How do we know that 1487 is a prime number? If 1487 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1487. Since 1487 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1487 is a prime number.

Other Facts about the Number 1487:

1487 is the difference of two squares:
744²-743² = 1487.

1487 is the third prime number in the fifth prime decade. See it in the list below: