# 1507 and Level 6

### Today’s Puzzle:

Level six puzzles are designed to be tricky, but if you examine the clues, there is a logical place to begin, and logic can help you complete the entire puzzle. ### Factors of 1507:

• 1507 is a composite number.
• Prime factorization: 1507 = 11 × 137
• 1507 has no exponents greater than 1 in its prime factorization, so √1507 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 1507 has exactly 4 factors.
• The factors of 1507 are outlined with their factor pair partners in the graphic below. ### More about the Number 1507:

1507 is the hypotenuse of a Pythagorean triple.
968-1155-1507

A divisibility trick tells us that all of the numbers in that triple are divisible by 11:
9 – 6 + 8 = 11,
1 – 1 + 5 – 5 = 0,
1 – 5 + 0 – 7 = -11.

Yes, you would need to understand negative numbers for that last one, but 11, 0, and -11 can all be evenly divided by 11 so the corresponding numbers are also divisible by 11.

In fact, 968-1155-1507 is just 11 times (88-105-137).

# 1495 and Level 6

### Today’s Puzzle:

Hint: The only way 12 can be put in the first column of this puzzle is to let one of the 60’s use it. We don’t have to know which 60 that is, to know that a 5 will go above that 60 in the top row. Knowing that, where does 5 have to go in the first column? That’s the logic needed to get started on this puzzle: ### Factors of 1495:

• 1495 is a composite number.
• Prime factorization: 1495 = 5 × 13 × 23.
• 1495 has no exponents greater than 1 in its prime factorization, so √1495 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 1495 has exactly 8 factors.
• The factors of 1495 are outlined with their factor pair partners in the graphic below. ### Other Facts about the Number 1495:

1495 is the hypotenuse of FOUR Pythagorean triples:
368-1449-1495, which is 23 times (16-63-65)
575-1380-1495, which is (5-12-13) times 115,
759-1288-1495, which is 23 times (33-56-65), and
897-1196-1495, which is (3-4-5) times 299.

# 1484 A Popsicle for a Hot Summer Day

### Today’s Puzzle:

It can be tricky to eat a melting popsicle on a hot summer day. Likewise, it can be tricky to solve a level 6 puzzle even if it looks like a popsicle!  Making it was my granddaughter’s idea. We hope you enjoy it! ### Factors of 1484:

If you know that 7 x 12 is 84, it isn’t hard to recognize that 1484 is divisible by 7.

• 1484 is a composite number.
• Prime factorization: 1484 = 2 × 2 × 7 × 53, which can be written 1484 = 2² × 7 × 53
• 1484 has at least one exponent greater than 1 in its prime factorization so √1484 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1484 = (√4)(√371) = 2√371
• 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 1484 has exactly 12 factors.
• The factors of 1484 are outlined with their factor pair partners in the graphic below. ### Other facts about the Number 1484:

1484 is the difference of two squares in two different ways:
372² – 370² = 1484
60² – 46² = 1484

784-1260-1484 which is 28 times (28-45-53)

# 1474 and Level 6

### Today’s Puzzle:

If you think that only kids will find the Find the Factors puzzles a challenge, then this puzzle will likely change your mind. Seriously, can you use logic to find where the factors from 1 to 12 must go to complete this puzzle? There is a reason this is a level 6 puzzle! ### Factors of 1474:

The last digit of 1474 is even so 1474 is divisible by 2. Yes, that was easy, but try this next one:
1 – 4 + 7 – 4 = 0, so 1474 can be evenly divided by 11.

• 1474 is a composite number.
• Prime factorization: 1474 = 2 × 11 × 67
• 1474 has no exponents greater than 1 in its prime factorization, so √1474 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 1474 has exactly 8 factors.
• The factors of 1474 are outlined with their factor pair partners in the graphic below. ### Another Fact about the Number 1474:

1474 can not be written as the sum of two squares or the difference of two squares, but it is still the short leg in two Pythagorean triples. Here is one of them:
1474-4368-4610, calculated from 2(67)(11), 67² – 11², 67² + 11²

# 1461 and Level 6

### Today’s Puzzle:

Look at all the clues in this level 6 puzzle. Can you deduce the logic needed to begin solving it? Deduce it correctly, then solving the rest of the puzzle will be a breeze. ### Factors of 1461:

• 1461 is a composite number.
• Prime factorization: 1461 = 3 × 487
• 1461 has no exponents greater than 1 in its prime factorization, so √1461 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 1461 has exactly 4 factors.
• The factors of 1461 are outlined with their factor pair partners in the graphic below. ### Another Fact about 1461:

1461 is the difference of two squares in two different ways:
731² – 730² = 1461
245² – 242² = 1461

# 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)