1740 I Think You’re Going to L♥ve This Multiplication Puzzle

Today’s Puzzle:

Here’s a heart-shaped level 2 puzzle for you to do. Just place all the numbers from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues.

Here’s the same puzzle in black and white:

Factors of 1740:

• 1740 is a composite number.
• Prime factorization: 1740 = 2 × 2 × 3 × 5 × 29, which can be written 1740 = 2² × 3 × 5 × 29.
• 1740 has at least one exponent greater than 1 in its prime factorization so √1740 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1740 = (√4)(√435) = 2√435.
• The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1740 has exactly 24 factors.
• The factors of 1740 are outlined with their factor pair partners in the graphic below.

1740 is the hypotenuse of FOUR Pythagorean triples:
204-1728-1740, which is 3 times (68-576-580),
288-1716-1740, which is 3 times (96-572-580),
1044-1392-1740, which is (3-4-5) times 348, and
1200-1260-1740, which is (20-21-29) times 60.

1740 is palindrome 606 in base 17 because
6(17²) + 0(17) + 6(1) = 1740.

1740 is the difference of two squares in four different ways:
436² – 434² = 1740,
148² – 142² = 1740,
92² – 82² = 1740, and
44² – 14² = 1740.

1731: The Sum of the Squares of Three Consecutive Prime Numbers

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 the numbers you write.

Factors of 1731:

• 1731 is a composite number.
• Prime factorization: 1731 = 3 × 577.
• 1731 has no exponents greater than 1 in its prime factorization, so √1731 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 1731 has exactly 4 factors.
• The factors of 1731 are outlined with their factor pair partners in the graphic below.

1731 is the hypotenuse of a Pythagorean triple:
144-1725-1731 which is 3 times (48-575-577).

OEIS.org informs us that 1731 is the sum of the squares of three consecutive prime numbers. Let’s find those three prime numbers. Since √(1731/3) rounds to 24, I’m guessing the middle prime number is 23. The prime numbers occurring before and after it are 19 and 29.

Is 19² + 23² + 29² = 1731? Yes, it is!

Here are some other ways that 1731 is the sum of three squares:
41² + 7² + 1² = 1731,
41² + 5² + 5² = 1731,
37² + 19² + 1² = 1731, and
29² + 29² + 7² = 1731.

1697 A Boot in the Window

Today’s Puzzle:

Tonight throughout many parts of the world children will place their polished boots in a window awaiting a visit from St. Nick. In the morning they will find their boots filled with favorite candies.

You can solve this boot puzzle by writing the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will become the start of a multiplication table.

Factors of 1697:

• 1697 is a prime number.
• Prime factorization: 1697 is prime.
• 1697 has no exponents greater than 1 in its prime factorization, so √1697 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1697 has exactly 2 factors.
• The factors of 1697 are outlined with their factor pair partners in the graphic below.

How do we know that 1697 is a prime number? If 1697 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1697. Since 1697 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1697 is a prime number.

1697 is the sum of two squares:
41² + 4² = 1697.

1697 is the hypotenuse of a Pythagorean triple:
328-1665-1697, calculated from 2(41)(4), 41² – 4², 41² + 4².

Here’s another way we know that 1697 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 41² + 4² = 1697 with 41 and 4 having no common prime factors, 1697 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1697. Since 1697 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1697 is a prime number.

1685 Oh, No! I’ve Created a Monster!

Today’s puzzle:

You may see some Frankenstein monsters walking about this time of year, but there’s no reason to be afraid of them or of this monster puzzle I’ve created. Simply write the numbers 1 to 12 in both factor areas so that the puzzle functions like a multiplication table.

Here’s the same puzzle without any added color, if that’s what you prefer:

Factors of 1685:

• 1685 is a composite number.
• Prime factorization: 1685 = 5 × 337.
• 1685 has no exponents greater than 1 in its prime factorization, so √1685 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 1685 has exactly 4 factors.
• The factors of 1685 are outlined with their factor pair partners in the graphic below.

1685 is the sum of two squares in two different ways:
41² + 2² = 1685, and
34² + 23² = 1685.

1685 is the hypotenuse of FOUR Pythagorean triples:
164-1677-1685, calculated from 2(41)(2), 41² – 2², 41² + 2²,
627-1564-1685, calculated from 34² – 23², 2(34)(23), 34² + 23²,
875-1440-1685, which is 5 times (175-288-337), and
1011-1348-1685, which is (3-4-5) times 337.

1674 and Level 2

Today’s Puzzle:

Make a multiplication table out of this puzzle. Can you see how to do it? The factors won’t be in the usual order, but I’m sure you can figure it out!

Factors of 1674:

• 1674 is a composite number.
• Prime factorization: 1674 = 2 × 3 × 3 × 3 × 31, which can be written 1674 = 2 × 3³ × 31.
• 1674 has at least one exponent greater than 1 in its prime factorization so √1674 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1674 = (√9)(√186) = 3√186.
• The exponents in the prime factorization are 1,3 and 1. Adding one to each exponent and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1674 has exactly 16 factors.
• The factors of 1674 are outlined with their factor pair partners in the graphic below.

From OEIS.org we learn that
1/1 + 1/2 + 1/3 + 1/4 + . . . + 1/1672 + 1/1673 ≈ 7.999888, but if you add the next tiny fraction, 1/1674, the sum will be a tiny bit more than 8 or approximately 8.000486.

That’s adding a whole lot of unit fractions just to get a sum over 8.

1664 and Level 2

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factor Cake for 1664:

We can make a factor cake for 1664 by doing some successive divisions. Divide 1664 by 2, divide that answer by 2, and so forth until you make a factor cake that looks like this:

Factors of 1664:

• 1664 is a composite number.
• Prime factorization: 1664 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 13, which can be written 1664 = 2⁷ × 13.
• 1664 has at least one exponent greater than 1 in its prime factorization so √1664 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1664 = (√64)(√26) = 8√26.
• The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1664 has exactly 16 factors.
• The factors of 1664 are outlined with their factor pair partners in the graphic below.

1664 is the sum of two squares:
40² + 8² = 1664.
That happened because it has a prime factor that leaves a remainder of 1 when divided by 4 AND all of its other prime factors are powers of 2 or perfect squares:

But that’s not all that cool about 1664. What patterns do you notice below?
2(24² + 16²) = 1664,
4(20² + 4²) = 1664,
8(12² + 8²) = 1664,
16(10² + 2²) = 1664,
32(6² + 4²) = 1664,
64(5² + 1²) = 1664, and
128(3² + 2²) = 1664.

1664 is the hypotenuse of a Pythagorean triple:
640-1536-1664, calculated from 2(40)(8), 40² – 8², 40² + 8².
That triple is also (5-12-13) times 128.

1651 Multiplication Fun

Today’s Puzzle:

Look how much fun these kids are having doing multiplication!

A game like that can help kids get ready to solve a fun puzzle based on the multiplication table.

Write each number from 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table.

Factors of 1651:

• 1651 is a composite number.
• Prime factorization: 1651 = 13 × 127.
• 1651 has no exponents greater than 1 in its prime factorization, so √1651 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 1651 has exactly 4 factors.
• The factors of 1651 are outlined with their factor pair partners in the graphic below.

1651 is the hypotenuse of a Pythagorean triple:
635-1524-1651, which is (5-12-13) times 127.

1651 is the 26th heptagonal number because
5(26²)/2 – 3(26)/2 = 1651.

1651 is a nice-looking palindrome in base 2:
1651₁₀ = 11001110011₂.
That just means that
2¹⁰ + 2⁹ + 2⁶ + 2⁵ + 2⁴+ 2¹+ 2⁰ = 1024 + 512 + 64 + 32 + 16 + 2 + 1 = 1651.

1640 A Level 2 Flower

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1640:

• 1640 is a composite number.
• Prime factorization: 1640 = 2 × 2 × 2 × 5 × 41, which can be written 1640 = 2³ × 5 × 41.
• 1640 has at least one exponent greater than 1 in its prime factorization so √1640 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1640 = (√4)(√410) = 2√410.
• 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 1640 has exactly 16 factors.
• The factors of 1640 are outlined with their factor pair partners in the graphic below.

Since 1640 = 40 x 41, we can be sure that 1640 is the sum of the first 40 even numbers.

1640 is the sum of two squares in two different ways:
38² + 14² = 1640, and
34² + 22² = 1640.

1640 is the hypotenuse of a Pythagorean triple in FOUR different ways:
360-1600-1640, which is 40 times (9-40-41),
672-1496-1640, calculated from 34² – 22², 2(34)(22), 34² + 22²,
but it is also 8 times (84-187-205),
984-1312-1640, which is (3-4-5) times 328, and
1064-1248-1640, calculated from 2(38)(14), 38² – 14², 38² + 14²,
but it is also 8 times (133-156-205).

1640₁₀ = 2222₉ because 2(9³ + 9² + 9¹ + 9⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.
1640₁₀ = 2020202₃ because 2(3⁶ + 3⁴ + 3² + 3⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.

1629 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work to make a multiplication table.

Factors of 1629:

1 + 6 + 2 = 9, so 1929 is divisible by both 3 and 9. (It’s only necessary to add the non-nine numbers together to check those two divisibility rules.)

• 1629 is a composite number.
• Prime factorization: 1629 = 3 × 3 × 181, which can be written 1629 = 3² × 181.
• 1629 has at least one exponent greater than 1 in its prime factorization so √1629 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1629 = (√9)(√181) = 3√181.
• 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 1629 has exactly 6 factors.
• The factors of 1629 are outlined with their factor pair partners in the graphic below.

1629 is the sum of two squares:
30² + 27² = 1629.

1629 is the hypotenuse of a Pythagorean triple:
171-1620-1629, calculated from 30² – 27², 2(30)(27), 30² + 27².
It is also 9 times (19-180-181).

1615 Should Today Be Root Ten Day?

Today’s Puzzle:

In a recent post, I compared π or (3.14…) with √10 or (3.16…). Steve Morris lives in England where today’s date is written 16-3, not 3-16. He jokingly commented, “So I guess Tuesday (16 March) should be Root Ten Day!” Seriously, day-month-year makes more sense as a writing convention than month-day-year.

Should today be Root Ten Day?
14 March has long been embraced as pi day in the United States, but should 16 March also be a quasi-holiday where kids eat roots like ten French fries or ten carrot sticks?

I remember one of my college professors telling his class that
√2 is about 1.4, and Valentines day is February 14,
√3 is about 1.7, and Saint Patrick’s day is March 17.

√1 is 1, and New Year’s Day is January 1, and
√10 is about 3.1, and Halloween is October 31. (I realize there is a rounding issue with that one.)

Oops. That could be said about all the fake holidays I’ve listed above.

And here’s a more serious thought:

Well, however you want to remember what √10 is or not, I decided to make today’s puzzle look like a square root sign for the fun of it. Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table.

Factors of 1615:

• 1615 is a composite number.
• Prime factorization: 1615 = 5 × 17 × 19.
• 1615 has no exponents greater than 1 in its prime factorization, so √1615 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 1615 has exactly 8 factors.
• The factors of 1615 are outlined with their factor pair partners in the graphic below.