# 1688 What Can Grow from a Little Seed?

### Today’s Puzzle:

Pumpkins are often harvested this time of year. It is amazing how one little seed properly planted and tended can grow into a vine that produces pumpkin after pumpkin after pumpkin.  Indeed great things can come from little things.

This level 5 puzzle starts out fairly easy. It doesn’t get very complicated until you’re about halfway through. Don’t give up! Your mind will grow as you use logic to find a way to work it out. Write the numbers from 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 1688:

• 1688 is a composite number.
• Prime factorization: 1688 = 2 × 2 × 2 × 211, which can be written 1688 = 2³ × 211.
• 1688 has at least one exponent greater than 1 in its prime factorization so √1688 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1688 = (√4)(√422) = 2√422.
• The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1688 has exactly 8 factors.
• The factors of 1688 are outlined with their factor pairs in the graphic below. ### More About the Number 1688:

1688 is the difference of two squares in two different ways:
423² – 421² = 1688, and
213² – 209² = 1688.

1688₁₀ = 888₁₄ because 8(14² + 14¹ + 14º) = 1688.

# 1678 and Level 5

### Today’s Puzzle:

Using logic, write the numbers from 1 to 10 in both the first column and the top row so that this puzzle will function like a multiplication table. ### Factors of 1678:

• 1678 is a composite number.
• Prime factorization: 1678 = 2 × 839.
• 1678 has no exponents greater than 1 in its prime factorization, so √1678 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 1678 has exactly 4 factors.
• The factors of 1678 are outlined with their factor pair partners in the graphic below. ### More about the number 1678:

Because 2(839)(1) = 1678, you can calculate the only Pythagorean triple that contains the number 1678:
The smaller leg will be 2(839)(1) = 1678.
The longer leg will be 839² – 1² = 703920.
The hypotenuse will be 839² + 1² = 703922.

# 1668 and Level 5

### 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 create a multiplication table. This is a level 5 puzzle, so some of the clues may be tricky. Remember to use logic and consider all possibilities for factors to avoid being tricked. ### Factors of 1668:

• 1668 is a composite number.
• Prime factorization: 1668 = 2 × 2 × 3 × 139, which can be written 1668 = 2² × 3 × 139.
• 1668 has at least one exponent greater than 1 in its prime factorization so √1668 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1668 = (√4)(√417) = 2√417.
• 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 1668 has exactly 12 factors.
• The factors of 1668 are outlined with their factor pair partners in the graphic below. ### More About the Number 1668:

1668 is the difference of two squares in two different ways:
418² – 416² = 1668, and
142² – 136² = 1668.

# 1655 and Level 5

### Today’s Puzzle:

Using logic, write all the numbers from 1 to 10 in both the first column and the top row of this puzzle so that those numbers are the factors of the given clues. ### Factors of 1655:

• 1655 is a composite number.
• Prime factorization: 1655 = 5 × 331.
• 1655 has no exponents greater than 1 in its prime factorization, so √1655 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 1655 has exactly 4 factors.
• The factors of 1655 are outlined with their factor pair partners in the graphic below. ### More About the Number 1655:

1655 is the hypotenuse of a Pythagorean triple:
993-1324-1655, which is (3-4-5) times 331.

# 1644 Level 5 Puzzles Are Not So Easy to Solve

### Today’s Puzzle:

This puzzle isn’t so easy to solve. For example, the common factor of clues 48 and 24 might be 4, 6, 8, or 12. Which one should you use? Logic will help answer that question. Give this puzzle a try? ### Factors of 1644:

1644 has 12 factors and is divisible by 12.

• 1644 is a composite number.
• Prime factorization: 1644 = 2 × 2 × 3 × 137, which can be written 1644 = 2² × 3 × 137.
• 1644 has at least one exponent greater than 1 in its prime factorization so √1644 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1644 = (√4)(√411) = 2√411.
• 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 1644 has exactly 12 factors.
• The factors of 1644 are outlined with their factor pair partners in the graphic below. ### More about the Number 1644:

1644 is the hypotenuse of a Pythagorean triple:
1056-1260-1644 which is 12 times (88-105-137).

# 1633 and Level 5

### Today’s Puzzle:

It might be tricky in a few places, but use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues behave like a multiplication table. ### Factors of 1633:

• 1633 is a composite number.
• Prime factorization: 1633 = 23 × 71.
• 1633 has no exponents greater than 1 in its prime factorization, so √1633 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 1633 has exactly 4 factors.
• The factors of 1633 are outlined with their factor pair partners in the graphic below. ### More about the Number 1633:

1633 is the difference of two squares in two different ways:
817² – 816² = 1633, and
47² – 24² = 1633.

### Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution. Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

### Factors of 1622:

• 1622 is a composite number.
• Prime factorization: 1622 = 2 × 811.
• 1622 has no exponents greater than 1 in its prime factorization, so √1622 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 1622 has exactly 4 factors.
• The factors of 1622 are outlined with their factor pair partners in the graphic below. ### More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

# 1619 A Pink Egg Hidden in the Grass

### Today’s Puzzle:

Easter is less than two weeks away. This pink puzzle is the first of three level-5 Easter eggs hidden amongst some blades of grass for you to find and solve. The puzzle might be a little tricky, but use logic every step of the way, and you’ll be able to find the unique solution: ### Factors of 1619:

• 1619 is a prime number.
• Prime factorization: 1619 is prime.
• 1619 has no exponents greater than 1 in its prime factorization, so √1619 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1619 has exactly 2 factors.
• The factors of 1619 are outlined with their factor pair partners in the graphic below.

How do we know that 1619 is a prime number? If 1619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1619. Since 1619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1619 is a prime number. ### More about the Number 1619:

1619 is the sum of two consecutive numbers:
809 + 810 = 1619.

1619 is also the difference of two consecutive squares:
810² – 809² = 1619.

What do you think about that?

# 1609 Pot of Gold

### 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! Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

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.

# 1599 and Level 5

### 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!