1559 We Need a Little Christmas Now

Today’s Puzzle:

2020 reminds me of another difficult year, 2001. In December of that year, Angela Lansbury sang We Need a Little Christmas Now with the Tabernacle Choir at Temple Square. I was able to watch the concert on television, and I remember the feeling the music brought me. What a wonderful gift music is! Yes, in 2020, we need a little Christmas now!

This level 2 puzzle brings a little Christmas now. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions as a type of multiplication table. I’m pretty sure you can figure it out!

Factors of 1559:

  • 1559 is a prime number.
  • Prime factorization: 1559 is prime.
  • 1559 has no exponents greater than 1 in its prime factorization, so √1559 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1559 has exactly 2 factors.
  • The factors of 1559 are outlined with their factor pair partners in the graphic below.

How do we know that 1559 is a prime number? If 1559 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1559. Since 1559 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1559 is a prime number.

More about the number 1559:

1559 is the sum of two consecutive numbers:
780 + 779 = 1559.

1559 is also the difference of two consecutive square numbers:
780² – 779² = 1559.

(Yes, I know, any odd whole number can make a similar claim.)

1558 The Reason for the Season

Today’s Puzzle:

What is the reason we have the Christmas season? The answer to that question actually begins with a cross…

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 numbers.

Factors of 1558:

  • 1558 is a composite number.
  • Prime factorization: 1558 = 2 × 19 × 41.
  • 1558 has no exponents greater than 1 in its prime factorization, so √1558 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 1558 has exactly 8 factors.
  • The factors of 1558 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1558:

1558 is the hypotenuse of a Pythagorean triple:
342-1520-1558, which is 38 times (9-40-41).

1557 Happy Birthday to My Brother, Andy!

Today’s puzzle:

Today is my brother Andy’s birthday. He enjoys solving puzzles so I made this 18 × 18 puzzle hoping that he will find it challenging.

This 18 × 18 multiplication table will be invaluable as you work to solve it. When you look for a clue in the table, its color will let you know how many times it appears in the table.

Print the puzzles or type the solution in this excel file: 10 Factors 1546-1557.

If you need a hint to solve the puzzle: One of the first things you will want to do is identify the clues that are multiples of 5, 10, or 15. Then use logic to determine which clues will use the two 5’s, the two 10’s, and the two 15’s.

Factors of 1557:

  • 1557 is a composite number.
  • Prime factorization: 1557 = 3 × 3 × 173, which can be written 1557 = 3² × 173
  • 1557 has at least one exponent greater than 1 in its prime factorization so √1557 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1557 = (√9)(√173) = 3√173
  • 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 1557 has exactly 6 factors.
  • The factors of 1557 are outlined with their factor pair partners in the graphic below.

More about the number 1557:

From OEIS.org we learn that 1557 has a rather fun square:
1557² = 2424249.

1557 is the sum of two squares:
39² + 6² = 1557.

1557 is the hypotenuse of a Pythagorean triple:
468-1485-1557, which is 9 times (52-165-173).
It can also be calculated from 2(39)(6), 39² – 6², 39² + 6².

1557 is also the difference of two squares three different ways:
779² – 778² = 1557,
261² – 258² = 1557, and
91² – 82² = 1557.

1556 Stacks Up Nicely!

What is special about the number 1556?

What makes 1556 stack up?

From OEIS.org we learn that
2² + 3² + 5² + 7² + 11² + 13² + 17² + 19² + 23² =  1556.
Yes, that’s the sum of the squares of the first nine prime numbers.
Those perfect squares can be stacked on top of each other as I illustrate in the graphic below:

Factors of 1556:

1556 (and every other whole number whose last two digits are 56) is divisible by 4:

  • 1556 is a composite number.
  • Prime factorization: 1556 = 2 × 2 × 389, which can be written 1556 = 2² × 389
  • 1556 has at least one exponent greater than 1 in its prime factorization so √1556 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1556 = (√4)(√389) = 2√389
  • 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 1556 has exactly 6 factors.
  • The factors of 1556 are outlined with their factor pair partners in the graphic below.

More about the Number 1556:

1556 is the sum of two squares:
34² + 20² = 1556.

1556 is the hypotenuse of a Pythagorean triple:
756 -1360-1556, which is 4 times (189-340-389).
It can also be calculated from 34² – 20², 2(34)(20), 34² + 20².

1556 is also the difference of two squares:
390²  – 388²  = 1556.

1555 Two Turkeys Too Tough To Try?

Today’s Puzzle:

Two turkeys too tough to try? That’s a six-word title made with alliteration and three homophones! It also describes the mystery-level turkey puzzles below. Those turkeys might look like identical twins at first glance, but if you look closely, you will see they are not quite the same.

Here are some questions to help you find a logical way to start either puzzle: Which two clues MUST use the two 6’s as factors? Are there any other clues that are multiples of 6? If so, what factors would those clues use?

Factors of 1555:

  • 1555 is a composite number.
  • Prime factorization: 1555 = 5 × 311.
  • 1555 has no exponents greater than 1 in its prime factorization, so √1555 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 1555 has exactly 4 factors.
  • The factors of 1555 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1555:

1555 is the hypotenuse of a Pythagorean triple:
933-1244-1555, which is (3-4-5) times 311.

1554 What Patterns Do You See?

Today’s Puzzle:

I like multiples of 111, including 1554. What cool patterns do you notice if a 2-digit number is multiplied by 111 as shown in the graphic below:

A Factor Tree for 1554:

Here’s a factor tree for 1554 that begins with the factor pair 14 × 111:

Factors of 1554:

  • 1554 is a composite number.
  • Prime factorization: 1554 = 2 × 3 × 7 × 37
  • 1554 has no exponents greater than 1 in its prime factorization, so √1554 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1554 has exactly 16 factors.
  • The factors of 1554 are outlined with their factor pair partners in the graphic below.

More about the Number 1554:

1554 is the hypotenuse of a Pythagorean triple:
504-1470-1554, which is (12-35-37) times 42.

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.

1552 Look for Clues around the Corner

Today’s Puzzle:

This puzzle has four sets of clues that turn the corner. You will need to look around those corners to solve it. Use logic and have fun!

Factors of 1552:

  • 1552 is a composite number.
  • Prime factorization: 1552 = 2 × 2 × 2 × 2 × 97, which can be written 1552 = 2⁴ × 97
  • 1552 has at least one exponent greater than 1 in its prime factorization so √1552 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1552 = (√16)(√97) = 4√97
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1552 has exactly 10 factors.
  • The factors of 1552 are outlined with their factor pair partners in the graphic below.

More about the Number 1552:

1552 is the sum of two squares:
36² + 16²  = 1552

1552 is the hypotenuse of a Pythagorean triple:
1040-1152-1552, calculated from 36² – 16², 2(36)(16), 36² + 16².
It is also 16 times (65-72-97).

OEIS.org looked around the corner at the two numbers preceding 1552 to find something special about that number: The sum of its prime factors equals the sum of the prime factors of those previous two numbers! That’s a cool enough fact that I decided to make this graphic:

1551 Two Straight Lines

Today’s Puzzle:

The clues in this level 4 puzzle form two straight lines, and most of the logic needed to solve it is rather straightforward. Can you find the factors from 1 to 10 without getting twisted up?

Factors of 1551:

1+5 = 6, a multiple of 3, so 1551 is divisible by 3.
Since 1551 is a palindrome with an even number of digits, it is also divisible by 11.

  • 1551 is a composite number.
  • Prime factorization: 1551 = 3 × 11 × 47.
  • 1551 has no exponents greater than 1 in its prime factorization, so √1551 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 1551 has exactly 8 factors.
  • The factors of 1551 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1551:

Did you notice how many of 1551’s factors are palindromes?

1551 is the difference of two squares in four different ways:
776² – 775² = 1551,
260² – 257² = 1551,
76² – 65² = 1551, and
40² – 7² = 1551. (Yes, we are just 7² or 49 numbers away from 1600, the next perfect square!)

1550 Lucky for You: Solving This Puzzle Is as Easy as Climbing Down a Ladder

Today’s Puzzle:

It’s Friday the 13th, so don’t walk under any ladders! Still, there isn’t any reason to avoid them entirely. Because this ladder puzzle is a level 3, the clues are given in a logical order to help you find the solution. Start at the top of the ladder, find the common factor of 10 and 18, then work your way down the ladder rung by rung, writing all the numbers from 1 to 10 in both the first column and the top row until you reach the bottom of the ladder.  Good Luck!

Here’s the same puzzle with no colors to distract you.

Factors of 1550:

  • 1550 is a composite number.
  • Prime factorization: 1550 = 2 × 5 × 5 × 31, which can be written 1550 = 2 × 5² × 31
  • 1550 has at least one exponent greater than 1 in its prime factorization so √1550 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1550 = (√25)(√62) = 5√62
  • 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 1550 has exactly 12 factors.
  • The factors of 1550 are outlined with their factor pair partners in the graphic below.

1550 Factor Tree:

Here’s one of a few possible different factor trees for 1550:

More about the Number 1550:

1550 is the hypotenuse of TWO Pythagorean triples:
434-1488-1550, which is (7-24-25) times 62, and
930-1240-1550, which is (3-4-5) times 310.