# 1439 and Level 6

You know the factors of 49, but to complete this puzzle, you will also have to determine the answers to a few questions:

Will the common factor of 24 and 32 be 4 or 8?
Will the common factor of 20 and 40 be 4, 5, or 10?
Will the common factor of 27 and 9 be 3 or 9?

Don’t guess and check! Look at the other clues, and some of the possibilities will be eliminated*. Once you’ve found the next clue to use, continue using logic until the whole puzzle is completed. Print the puzzles or type the solution in this excel file: 10 Factors 1432-1442

• 1439 is a prime number.
• Prime factorization: 1439 is prime.
• 1439 has no exponents greater than 1 in its prime factorization, so √1439 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1439 has exactly 2 factors.
• The factors of 1439 are outlined with their factor pair partners in the graphic below. How do we know that 1439 is a prime number? If 1439 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1439. Since 1439 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1439 is a prime number.

* Did you notice that either one of the 24’s or 6 must use a 3 because only two 6s are allowed to be used as factors? What does that tell you about the common factor of 27 and 9?

# 1424 Logic and Fear

Should this puzzle excite fear in you or anyone else? The logical answer is no, of course not. Logic can also help you to find the solution, so don’t be afraid, give it a try! Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Finding the factors of 1424 shouldn’t bring fear to you either:

• 1424 is a composite number.
• Prime factorization: 1424 = 2 × 2 × 2 × 2 × 89, which can be written 1424 = 2⁴ × 89
• 1424 has at least one exponent greater than 1 in its prime factorization so √1424 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1424 = (√16)(√89) = 4√89
• 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 1424 has exactly 10 factors.
• The factors of 1424 are outlined with their factor pair partners in the graphic below. 1424 is the sum of two squares:
32² + 20² = 1424

1424 is the hypotenuse of a Pythagorean triple:
624-1280-1424 calculated from 32² – 20², 2(32)(20), 32² + 20²

# 1415 and Level 6

Very likely when you look at this puzzle common factors of 40 and 10, 8 and 16, 9 and 18, and 20 and 40 will pop into your head. Will they be the right common factors that work with all the other clues in the puzzle to produce a unique solution? Let logic be your guide when finding the factors. Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll tell you something about the puzzle number, 1415:

• 1415 is a composite number.
• Prime factorization: 1415 = 5 × 283
• 1415 has no exponents greater than 1 in its prime factorization, so √1415 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 1415 has exactly 4 factors.
• The factors of 1415 are outlined with their factor pair partners in the graphic below. 1415 is also the hypotenuse of a Pythagorean triple:
849-1132-1415 which is (3-4-5) times 283.

# 1399 and Level 6

The eligible common factors of 48 and 72 are 6, 8, and 12. The common factors for 10 and 30 are 5 and 10.  Don’t guess and check the possibilities! Can you figure out the logic needed to start this puzzle? Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s a little information about the number 1399:

• 1399 is a prime number.
• Prime factorization: 1399 is prime.
• 1399 has no exponents greater than 1 in its prime factorization, so √1399 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1399 has exactly 2 factors.
• The factors of 1399 are outlined with their factor pair partners in the graphic below. How do we know that 1399 is a prime number? If 1399 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1399. Since 1399 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1399 is a prime number.

1399 is the difference of two squares:
700² – 699² = 1399

# 1384 and Level 6

Guessing and checking can be so frustrating! If instead, you study the clues to find a logical place to start this puzzle, you are more likely to be able to find the one and only solution. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the number 1384:

• 1384 is a composite number.
• Prime factorization: 1384 = 2 × 2 × 2 × 173, which can be written 1384 = 2³ × 173
• 1384 has at least one exponent greater than 1 in its prime factorization so √1384 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1384 = (√4)(√346) = 2√346
• 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 1384 has exactly 8 factors.
• The factors of 1384 are outlined with their factor pairs in the graphic below. 1384 is the hypotenuse of a Pythagorean triple:
416-1320-1384 which is 8 times (52-165-173)

# 1363 and Level 6

The common factors of 60 and 30 allowed in the puzzle are 5, 6, and 10. Which one is the logical choice? Look at the other clues in the puzzle and you should be able to eliminate two of the choices. Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few thoughts about the puzzle number, 1363:

• 1363 is a composite number.
• Prime factorization: 1363 = 29 × 47
• 1363 has no exponents greater than 1 in its prime factorization, so √1363 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 1363 has exactly 4 factors.
• The factors of 1363 are outlined with their factor pair partners in the graphic below. 1363 is the hypotenuse of a Pythagorean triple:
940-987-1363 which is (20-21-29) times 47

# 1355 I Can Solve This Puzzle, Can You?

Level 6 puzzles are a little tricky because there is more than one possible common factor for every set of clues on the same row or column.

You can still solve it using logic. I can solve it. Can you? Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you need some logical hints, the video below will be helpful:

Now I’ll share some information about the puzzle number, 1355:

• 1355 is a composite number.
• Prime factorization: 1355 = 5 × 271
• 1355 has no exponents greater than 1 in its prime factorization, so √1355 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 1355 has exactly 4 factors.
• The factors of 1355 are outlined with their factor pair partners in the graphic below. 1355 is the hypotenuse of a Pythagorean triple:
813-1084-1355 which is (3-4-5) times 271

# 1339 Written Eligible Factor Pairs Can Reveal the Needed Logic

The logic needed for this particular level 6 puzzle is VERY complicated so I wrote out the eligible factor pairs of all the clues for you. Or you can start from scratch: Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here are a few facts about the puzzle number, 1339:

• 1339 is a composite number.
• Prime factorization: 1339 = 13 × 103
• 1339 has no exponents greater than 1 in its prime factorization, so √1339 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 1339 has exactly 4 factors.
• The factors of 1339 are outlined with their factor pairs in the graphic below. 1339 is the hypotenuse of a Pythagorean triple:
515-1236-1339 which is (5-12-13) times 103

# 1329 Flight Plans

Many people fly home or away from home for the holidays. Here’s a puzzle to occupy some of your time while you’re in flight.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll write a little bit about the number 1329:

• 1329 is a composite number.
• Prime factorization: 1329 = 3 × 443
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1329 has exactly 4 factors.
• Factors of 1329: 1, 3, 443, 1329
• Factor pairs: 1329 = 1 × 1329 or 3 × 443
• 1329 has no square factors that allow its square root to be simplified. √1329 ≈ 36.45545

1329 is divisible by 3 because it is made with three consecutive numbers (1, 2, 3) and 9, a number divisible by 3.

1329 looks interesting in some other bases:
It’s 929 in BASE 12,
369 in BASE 20
234 in BASE 25

# 1320 Christmas Factor Trees

Regardless of its size, an evergreen tree is a mighty symbol at Christmastime. Today’s factoring puzzle features a couple of relatively small Christmas trees, but don’t think for even one minute that these little trees make for an easy puzzle. It’s a level 6 puzzle so there are several places that the clues might trick you. Use logic through the entire process, and you should be able to solve it!

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

That factoring puzzle has nothing to do with the factors of 1320. In case you are looking for factor trees for the number 1320, here are a few of the MANY possible ones:

Here are some facts about the number 1320:

• 1320 is a composite number.
• Prime factorization: 1320 = 2 × 2 × 2 × 3 × 5 × 11, which can be written 1320 = 2³ × 3 × 5 × 11.
• The exponents in the prime factorization are 3, 1, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) )(1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 1320 has exactly 32 factors.
• Factors of 1320: 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 20, 22, 24, 30, 33, 40, 44, 55, 60, 66, 88, 110, 120, 132, 165, 220, 264, 330, 440, 660, 1320
• Factor pairs: 1320 = 1 × 1320, 2 × 660, 3 × 440, 4 × 330, 5 × 264, 6 × 220, 8 × 165, 10 × 132, 11 × 120, 12 × 110, 15 × 88, 20 × 66, 22 × 60, 24 × 55, 30 × 44 or 33 × 40
• Taking the factor pair with the largest square number factor, we get √1320 = (√4)(√330) = 2√330 ≈ 36.331804

1320 is the sum of consecutive primes FOUR different ways:
It is the sum of the eighteen primes from 37 to 109,
107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1320,
149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 = 1320, and
659 + 661 = 1320

1320 is the hypotenuse of a Pythagorean triple:
792-1056-1320 which is (3-4-5) times 264