Puzzle

1609 Pot of Gold

/ ivasallay / Leave a comment

Today’s Puzzle:

They say at the end of the rainbow, there is a pot of gold that belongs to some leprechaun. Because this is a Find the Factors 1 to 14 puzzle, this pot of gold has some choice mathematical nuggets. For example, is 7 or 14 the common factor of 70 and 84? Don’t guess which one is the common factor for the puzzle. Use logic to eliminate one of those possibilities instead.

Likewise, both 6 and 9 are common factors of 18 and 54. And 4, 5, and 10 are all common factors of 20 and 40. Logic will narrow each possibility down to one possible factor!

I like that you also need to find the common factor of 126 and 36. I noticed a pattern with those clues. The pattern is limited to the multiplication facts given below, but I think it is still a pretty cool pattern.

Here’s the pattern I saw for 126 and 36:

  • Since 9 is one of the factors, the sum of the digits of any of the products equals 9.
  • 1 + 2 = 3. The sum of the first two numbers of the product in the first column equals the first part of the product in the second column.
  • Obviously, both clues end in 6 so the last digit of their other factors will end with the same number, 4.

That should give you a good start in solving the puzzle!

Factors of 1609:

  • 1609 is a prime number.
  • Prime factorization: 1609 is prime.
  • 1609 has no exponents greater than 1 in its prime factorization, so √1609 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1609 has exactly 2 factors.
  • The factors of 1609 are outlined with their factor pair partners in the graphic below.

How do we know that 1609 is a prime number? If 1609 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1609. Since 1609 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1609 is a prime number.

More about the Number 1609:

1609² = 2588881. That’s a perfect square, followed by four 8’s, followed by a perfect square. OEIS.org reports that 1609² is the smallest perfect square with four 8’s in a row.

1609 is the sum of two squares:
40² + 3² = 1609.

1609 is the hypotenuse of a Pythagorean triple:
240-1591-1609, calculated from 2(40)(3), 40² – 3², 40² + 3².

Here’s another way we know that 1609 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 3² = 1609 with 40 and 3 having no common prime factors, 1609 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1609. Since 1609 is not divisible by 5, 13, 17, 29, or 37, we know that 1609 is a prime number.

 

1608 Rainbow

/ ivasallay / Leave a comment

Today’s Puzzle:

We often think of rainbows around Saint Patrick’s Day. Here is a rainbow puzzle for you to solve. It won’t be all that easy even if I tell you that
13 × 14 = 182,
12 × 13 = 156, and
8 ×  14 = 112.

Good luck!

If you’d like to print the puzzle but not use so much ink, here’s a puzzle with all the same clues:

 

Factor Rainbow for 1608:

The number 1608 has enough factors to make an impressive factor rainbow:

Factors of 1608:

  • 1608 is a composite number.
  • Prime factorization: 1608 = 2 × 2 × 2 × 3 × 67, which can be written 1608 = 2³ × 3 × 67
  • 1608 has at least one exponent greater than 1 in its prime factorization so √1608 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1608 = (√4)(√402) = 2√402
  • 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 1608 has exactly 16 factors.
  • The factors of 1608 are outlined with their factor pair partners in the graphic below.

More about the Number 1608:

1608 is the difference of two squares in four different ways:
403² – 401² = 1608,
203² – 199² = 1608,
137² – 131² = 1608, and
73² – 61² = 1608.

1607 Shillelagh

/ ivasallay / Leave a comment

Today’s Puzzle:

A Shillelagh is an Irish wooden walking stick. This Shillelagh is keeping with our Saint Patrick’s Day theme, but it is a Find the Factors 1 to 14 puzzle.  Brutal! It will be a whole lot less tricky for you to solve because I made it a level 3 puzzle: The logic needed to solve the puzzle is built in. Just start with the clue at the top of the puzzle and work your way down cell by cell until you have found all the factors. So crack on!

Factors of 1607:

  • 1607 is a prime number.
  • Prime factorization: 1607 is prime.
  • 1607 has no exponents greater than 1 in its prime factorization, so √1607 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1607 has exactly 2 factors.
  • The factors of 1607 are outlined with their factor pair partners in the graphic below.

How do we know that 1607 is a prime number? If 1607 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1607. Since 1607 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1607 is a prime number.

More about the Number 1607:

1607 is the sum of two consecutive numbers:
803 + 804 = 1607.

1607 is also the difference of two consecutive numbers:
804² – 803² = 1607.

Did you notice what happened there? Try this next one:

1607² = 2582449.

1607²/2 = 1291224.5.

(1607-1291224-1291225) is a primitive Pythagorean triple.

Cool, isn’t it?

1606 A Lucky Shamrock

/ ivasallay / Leave a comment

Today’s Puzzle:

Even if you don’t know all the factors of some of these clues in this shamrock puzzle, there are enough that you will know, and you can figure the rest out easily. Lucky you!

Factors of 1606:

  • 1606 is a composite number.
  • Prime factorization: 1606 = 2 × 11 × 73.
  • 1606 has no exponents greater than 1 in its prime factorization, so √1606 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 1606 has exactly 8 factors.
  • The factors of 1606 are outlined with their factor pair partners in the graphic below.

More about the Number 1606:

1606 is the hypotenuse of a Pythagorean triple:
1056-1210-1606. which is 22 times (48-55-73).

1606 is in a couple of other Pythagorean triples that can be calculated from
2(803)(1), 803² – 1², 803² + 1², and
2(73)(11), 73² – 11², 73² + 11².

1605 and Another Big Level One

/ ivasallay / Leave a comment

Today’s Puzzle:

Many of the clues in this puzzle do not appear in a 1 to 10 puzzle or a 1 to 12 puzzle. Can you find all the factors anyway? Solving this puzzle can help you solve any other Find the Factors 1 -14 puzzles.

Factors of 1605:

  • 1605 is a composite number.
  • Prime factorization: 1605 = 3 × 5 × 107.
  • 1605 has no exponents greater than 1 in its prime factorization, so √1605 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 1605 has exactly 8 factors.
  • The factors of 1605 are outlined with their factor pair partners in the graphic below.

More about the Number 1605:

1605 is the hypotenuse of a Pythagorean triple:
963-1284-1605, which is (3-4-5) times 321.

1605 is the difference of two squares four different ways:
803² – 802² = 1605,
269² – 266² = 1605,
163² – 158² = 1605, and
61² – 46² = 1605.

1604 and a Big Level One

/ ivasallay / Leave a comment

Today’s Puzzle:

1604 was my house number most of my childhood. Perhaps this number is making me feel a little more playful. I decided my next few puzzles would be based on a 1 to 14 multiplication table. I find that I personally use clues like 26, 39, 52, 65, 78, 104, 143, and 169 often in life. For example, why are there 52 playing cards in a deck of cards? To help you get more familiar with the 1 to14 multiplication table, we will start with this level one puzzle:

Factors of 1604:

  • 1604 is a composite number.
  • Prime factorization: 1604 = 2 × 2 × 401, which can be written 1604 = 2² × 401
  • 1604 has at least one exponent greater than 1 in its prime factorization so √1604 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1604 = (√4)(√401) = 2√401
  • 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 1604 has exactly 6 factors.
  • The factors of 1604 are outlined with their factor pair partners in the graphic below.

More about the Number 1604:

1604 is the sum of two squares:
40² + 2² = 1604.

1604 is the hypotenuse of a Pythagorean triple:
160-1596-1604, calculated from 2(40)(2), 40² – 2², 40² + 2².
It is also 4 times (40-399-401).

1603 Challenge Puzzle

/ ivasallay / Leave a comment

Today’s Puzzle:

Use logic to write the numbers 1 to 10 in each of the four boldly outlined areas so that the puzzle functions like four multiplication tables that work together.

Print the puzzles or type the solution in this excel file: 10 Factors 1594-1603.

Factors of 1603:

  • 1603 is a composite number.
  • Prime factorization: 1603 = 7 × 229.
  • 1603 has no exponents greater than 1 in its prime factorization, so √1603 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 1603 has exactly 4 factors.
  • The factors of 1603 are outlined with their factor pair partners in the graphic below.

More about the Number 1603:

1603 is the hypotenuse of a Pythagorean triple:
420-1547-1603, which is 7 times (60-221-229).

1602 Mystery

/ ivasallay / Leave a comment

Today’s Puzzle:

Is the logic needed to solve this puzzle simple or complicated? That question is part of the mystery!

Factors of 1602:

  • 1602 is a composite number.
  • Prime factorization: 1602 = 2 × 3 × 3 × 89, which can be written 1602 = 2 × 3² × 89.
  • 1602 has at least one exponent greater than 1 in its prime factorization so √1602 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1602 = (√9)(√178) = 3√
  • 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 1602 has exactly 12 factors.
  • The factors of 1602 are outlined with their factor pair partners in the graphic below.

More about the Number 1602:

How is 1602 the sum of two squares?
39² + 9² = 1602.

How is 1602 the hypotenuse of a Pythagorean triple?
702-1440-1602, calculated from 2(39)(9), 39² – 9², 39² + 9².
That triple is also 9 times (78-160-178).

1601 and Level 6

/ ivasallay / Leave a comment

Today’s Puzzle:

Remember to use logic for EVERY step when solving this puzzle. Guessing and checking will likely just frustrate you! It’s a level 6 puzzle, so it could be tricky.
Keep in mind:
1 and 2 are common factors of 6 and 8,
3 and 9 are common factors of 9 and 27,
4 & 8 are common factors of 32 and 8, and
4, 5, & 10 are common factors of 20 & 40.

As always, there is only one solution.

Factors of 1601:

  • 1601 is a prime number.
  • Prime factorization: 1601 is prime.
  • 1601 has no exponents greater than 1 in its prime factorization, so √1601 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1601 has exactly 2 factors.
  • The factors of 1601 are outlined with their factor pair partners in the graphic below.

How do we know that 1601 is a prime number? If 1601 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1601. Since 1601 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1601 is a prime number.

More about the Number 1601:

1601 is one more than a perfect square, so it is the sum of two squares:
40² + 1² = 1601.

1601 is the hypotenuse of a Pythagorean triple:
80-1599-1601, calculated from 2(40)(1), 40² – 1², 40² + 1².

Here’s another way we know that 1601 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 40² + 1² = 1601 with 40 and 1 having no common prime factors, 1601 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1601. Since 1601 is not divisible by 5, 13, 17, 29, or 37, we know that 1601 is a prime number.

1599 and Level 5

/ ivasallay / Leave a comment

Today’s Puzzle:

Write the numbers 1 to 10 in both the first column and the top row so that this level 5 puzzle will function like a multiplication table. Use logic with every step.

Factors of 1599:

  • 1599 is a composite number.
  • Prime factorization: 1599 = 3 × 13 × 41.
  • 1599 has no exponents greater than 1 in its prime factorization, so √1599 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 1599 has exactly 8 factors.
  • The factors of 1599 are outlined with their factor pair partners in the graphic below.

More about the Number 1599:

1599 is the hypotenuse of FOUR Pythagorean triples:
276-1575-1599, which is 3 times (92-525-533),
351-1560-1599, which is 39 times (9-40-41),
615-1476-1599, which is (5-12-13) times 123, and
924-1305-1599, which is 3 times (308-435-533).

1599 is the difference of two squares four different ways:
800² – 799² = 1599,
268² – 265² = 1599,
68² – 55² = 1599, and
40² – 1² = 1599.
Yes, we are just one number away from a perfect square!