Puzzles

1713 A Little More of a Subtraction Distraction

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Today’s Puzzle:

It occurred to me that as long as the last box is neither 1 nor 12 that I could leave the clue above it blank. Can you still solve the puzzle?

Factors of 1713:

  • 1713 is a composite number.
  • Prime factorization: 1713 = 3 × 571.
  • 1713 has no exponents greater than 1 in its prime factorization, so √1713 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 1713 has exactly 4 factors.
  • The factors of 1713 are outlined with their factor pair partners in the graphic below.

More About the Number 1713:

1713 is the difference of two squares in two different ways:
857² – 856² = 1713, and
287² – 284² = 1713.

1713 is the sum of two, three, and six consecutive numbers:
856 + 857 = 1713,
570 + 571 + 572 = 1713, and
283 + 284 + 285 + 286 + 287 + 288 = 1713.

Do you see any relationship between those two facts?

1712 Can You Make the Factors Fit?

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Today’s Puzzle:

This Factor Fits puzzle starts off fairly easy before it potentially might give you fits trying to place the rest of the factors. Are you game?

Factors of 1712:

  • 1712 is a composite number.
  • Prime factorization: 1712 = 2 × 2 × 2 × 2 × 107, which can be written 1712 = 2⁴ × 107.
  • 1712 has at least one exponent greater than 1 in its prime factorization so √1712 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1712 = (√16)(√107) = 4√107.
  • 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 1712 has exactly 10 factors.
  • The factors of 1712 are outlined with their factor pair partners in the graphic below.

More About the Number 1712:

1712 is the difference of two squares in three different ways:
429² – 427² = 1712,
216² – 212² = 1712, and
111² – 103² = 1712.

1709 Sometimes “Guess and Check” Is a Good Strategy

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Today’s Puzzle:

Most of the puzzles I publish are logic puzzles, and I encourage you to find the logic of the puzzle and not guess and check.  However, guess and check is a legitimate strategy in mathematics, and it is a legitimate strategy to solve this particular puzzle.

Since one of the clues is -9, we know that the two boxes under it must be [1, 10], [2, 11], or [3, 12].

Suppose you assume it’s 1 – 10 = -9. If you fill out the rest of the boxes you would get:

You know that isn’t right because zero is not a number from 1 to 12. No problem. Simply add one to each of the numbers you wrote in, and the puzzle will be solved with only numbers from 1 to 12.

Suppose you assumed it’s 3 -12 = -9. The rest of the boxes would look like this:

Again, 13 is not included in the numbers from 1 to 12, but you can fix it by subtracting 1 from each of the numbers you wrote in. Easy Peasy.

Factors of 1709:

  • 1709 is a prime number.
  • Prime factorization: 1709 is prime.
  • 1709 has no exponents greater than 1 in its prime factorization, so √1709 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1709 has exactly 2 factors.
  • The factors of 1709 are outlined with their factor pair partners in the graphic below.

How do we know that 1709 is a prime number? If 1709 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1709. Since 1709 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1709 is a prime number.

More About the Number 1709:

1709 is the sum of two squares:
35² + 22² = 1709.

1709 is the hypotenuse of a Pythagorean triple:
741-1540-1709, calculated from 35² – 22², 2(35)(22), 35² + 22².

Here’s another way we know that 1709 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 22² = 1709 with 35 and 22 having no common prime factors, 1709 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1709. Since 1709 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1709 is a prime number.

1708 Happy Birthday, Jo Morgan!

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Today’s Puzzle:

A few days ago I published a new kind of factoring puzzle. Jo Morgan of Resourceaholic.com, keeps an eye out for new mathematical resources on Twitter. She was one of the first to notice and like my puzzle. Because of her, lots of other people noticed the puzzle, too. Today is Jo’s birthday, and I decided to make a similar puzzle for her to enjoy. You might find it slightly more difficult than the earlier puzzle, but use logic from the beginning, and you will be able to fit in all the factors.

Factors of 1708:

  • 1708 is a composite number.
  • Prime factorization: 1708 = 2 × 2 × 7 × 61, which can be written 1708 = 2² × 7 × 61.
  • 1708 has at least one exponent greater than 1 in its prime factorization so √1708 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1708 = (√4)(√427) = 2√427.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1708 has exactly 12 factors.
  • The factors of 1708 are outlined with their factor pair partners in the graphic below.

More About the Number 1708:

1708 is the hypotenuse of a Pythagorean triple:
308-1680-1708, which is 28 times (11-60-61).

1707 Subtraction Distraction

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Today’s Puzzle:

Even though this Subtraction Distraction puzzle has more boxes than the one I published a couple of weeks ago, it is actually an easier puzzle. Can you write the numbers 1 to 12 in the boxes so that each triangle is its adjacent left box minus its adjacent right box?

Factors of 1707:

1 + 7 + 0 + 7 = 15, a multiple of 3, so 1707 is divisible by 3.

  • 1707 is a composite number.
  • Prime factorization: 1707 = 3 × 569.
  • 1707 has no exponents greater than 1 in its prime factorization, so √1707 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 1707 has exactly 4 factors.
  • The factors of 1707 are outlined with their factor pair partners in the graphic below.

More About the Number 1707:

1707 is the hypotenuse of a Pythagorean triple:
693-1560-1707, which is 3 times (231-520-569).

1704 Christmas Factor Tree

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Today’s Puzzle:

If you know the factors of the clues in this Christmas tree, and you use logic, it is possible to write each number from 1 to 12 in both the first column and the top row to make a multiplication table. It’s a level six puzzle, so it won’t be easy, even for adults, but can YOU do it?

Factors of 1704:

If you were expecting to see a factor tree for the number 1704, here is one of several possibilities:

  • 1704 is a composite number.
  • Prime factorization: 1704 = 2 × 2 × 2 × 3 × 71, which can be written 1704 = 2³ × 3 × 71.
  • 1704 has at least one exponent greater than 1 in its prime factorization so √1704 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1704 = (√4)(√426) = 2√426.
  • 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 1704 has exactly 16 factors.
  • The factors of 1704 are outlined with their factor pair partners in the graphic below.


More About the Number 1704:

1704 is the difference of two squares in FOUR different ways:
427² – 425² = 1704,
215² – 211² = 1704,
145² – 139² = 1704, and
77² – 65² = 1704.

Why was Six afraid of Seven? Because Seven ate Nine.
1704 is 789 in a different base:
1704₁₀ = 789₁₅ because 7(15²) + 8(15¹) + 9(15º) = 1704.

1699 Sweet Candy Cane

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Today’s Puzzle:

Solving this candy cane puzzle can be a sweet experience. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Factors of 1699:

  • 1699 is a prime number.
  • Prime factorization: 1699 is prime.
  • 1699 has no exponents greater than 1 in its prime factorization, so √1699 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1699 has exactly 2 factors.
  • The factors of 1699 are outlined with their factor pair partners in the graphic below.

How do we know that 1699 is a prime number? If 1699 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1699. Since 1699 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1699 is a prime number.

More About the Number 1699:

1699 is the third prime in a prime triple. What are the other two primes in that triple?

1699 is the difference of two squares:
850² – 849² = 1699.

1698 A Little Virgács and Candy

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Today’s Puzzle:

If you were a child in Hungary, you might have found some virgács and some candy in your boot this morning. Mikulás (St. Nick) would have given you the candy because of how good you’ve been this year, and the virgács for those times you weren’t so good.

This virgács and candy puzzle is like a mixed-up multiplication table. It is a lot easier to solve because I made it a level 3 puzzle. First, find the common factor of 56 and 72 that will allow only numbers between 1 and 12 to go in the first column. Put the factors in the appropriate cells, then work your way down the puzzle, row by row until each number from 1 to 12 is in both the first column and the top row.

Factors of 1698:

  • 1698 is a composite number.
  • Prime factorization: 1698 = 2 × 3 × 283.
  • 1698 has no exponents greater than 1 in its prime factorization, so √1698 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 1698 has exactly 8 factors.
  • The factors of 1698 are outlined with their factor pair partners in the graphic below.

More About the Number 1698:

1698 = 2(849)(1), so it is a leg in the Pythagorean triple calculated from
2(849)(1), 849² – 1², 849² + 1².

1698 = 2(283)(3), so it is a leg in the Pythagorean triple calculated from
2(283)(3), 283² – 3², 283² + 3².

1697 A Boot in the Window

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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.

More About the Number 1697:

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.

1696 Inverses

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Today’s Puzzle:

Joseph Nebus of Nebusresearch recently wrote an extensive post about inverses at my request. Although inverses are often fascinating, advanced topics in mathematics, they can also be quite simple. For example, solving this puzzle will involve using the inverse of multiplication, division, AND it is the simplest division possible this time. Since it is December, I made this puzzle look like a gift for you. Just write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues could become a multiplication table.

For any gift you receive, you can invoke inverse exploring questions such as “how was this gift made?”

If you want to print the puzzle without color, here it is:

Factors of 1696:

  • 1696 is a composite number.
  • Prime factorization: 1696 = 2 × 2 × 2 × 2 × 2 × 53, which can be written 1696 = 2⁵ × 53.
  • 1696 has at least one exponent greater than 1 in its prime factorization so √1696 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1696 = (√16)(√106) = 4√106.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1696 has exactly 12 factors.
  • The factors of 1696 are outlined with their factor pair partners in the graphic below.

More About the Number 1696:

1696 is the sum of two squares:
36² + 20² = 1696.

1696 is also the hypotenuse of a Pythagorean triple:
896-1440 -1696, which is 32 times (28-45-52).
It can also be calculated from 36² – 20², 2(36)(20), 36² + 20².

1696 is also the difference of two squares in FOUR different ways:
425² – 423² = 1696,
214² – 210² = 1696,
110² – 102² = 1696, and
61² – 45² = 1696.
I found those equations by using the even factor pairs of 1696 and taking the inverse of the fact that a² – b² = (a + b)(a – b).