1515 and Level 3

Today’s Puzzle:

Start with the greatest common factor of 30 and 48, write the factors in the appropriate boxes, then work your way down this level 3 puzzle row by row using logic until you have found all the factors. You can do this!

Factors of 1515:

  • 1515 is a composite number.
  • Prime factorization: 1515 = 3 × 5 × 101.
  • 1515 has no exponents greater than 1 in its prime factorization, so √1515 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 1515 has exactly 8 factors.
  • The factors of 1515 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1515:

1515 is the hypotenuse of FOUR Pythagorean triples:
300-1485-1515, which is 15 times (20-99-101),
651-1368-1515, which is 3 times (217-456-505),
909-1212-1515, which is (3-4-5) times 303,
1008-1131-1515, which is 3 times (336-377-505).

1504 and Level 3

Today’s Puzzle:

Since this is a level 3 puzzle the clues are given in a logical order from top to bottom. Write the factors 1 to 10 in the first column and again in the top row.

Usually, you only have to consider the previous clues when finding the factors in a level 3 puzzle, but when you consider if 4 = 2 × 2 or 1 × 4, you will also have to look at a clue below it. You can do this!

Factors of 1504:

  • 1504 is a composite number.
  • Prime factorization: 1504 = 2 × 2 × 2 × 2 × 2 × 47, which can be written 1504 = 2⁵ × 47
  • 1504 has at least one exponent greater than 1 in its prime factorization so √1504 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1504 = (√16)(√94) = 4√94
  • 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 1504 has exactly 12 factors.
  • The factors of 1504 are outlined with their factor pair partners in the graphic below.

More about the Number 1504:

1504 is the difference of two squares in four different ways:
377² – 375² = 1504
190² – 186² = 1504
98² – 90² = 1504
55² – 39² = 1504

1492 The Niña, the Pinta, and the Santa Maria

Today’s Puzzle:

I debated with myself about whether or not I should mention the famous poem about 1492 in this post. Then I looked at the puzzle I had already created for today and noticed it would not be too much of a stretch to say the twelve clues look a little bit like three ships.  So whether or not I should, I decided to go ahead and mention the Niña, the Pinta, and the larger Santa Maria in the title. The ships themselves are innocent of any barbaric acts and might just be the three most famous ships in world history.

Start at the top of this level 3 puzzle and work your way down cell by cell using logic until you have written all the factors from 1 to 12 in both the first column and the top row.

Here’s the same puzzle without color:

Factors of 1492:

  • 1492 is a composite number.
  • Prime factorization: 1492 = 2 × 2 × 373, which can be written 1492 = 2² × 373
  • 1492 has at least one exponent greater than 1 in its prime factorization so √1492 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1492 = (√4)(√373) = 2√373
  • 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 1492 has exactly 6 factors.
  • The factors of 1492 are outlined with their factor pair partners in the graphic below.

Other Facts About the Number 1492:

1492 is the sum of two squares:
36² + 14² = 1492

1492 is the hypotenuse of a Pythagorean triple:
1008-1100-1492, which is 4 times (252-275-373)
and can also be calculated from 2(36)(14), 36² – 14², 36² + 14².

1480 and Level 3

Today’s Puzzle:

This is a level 3 puzzle. Start with the common factor of 21 and 14, then fill in the cells in the first column in order from top to bottom as well as the factors in the top row until you have placed all of the factors from 1 to 10 in both of those places.

Factors of 1480:

80 is divisible by 8 and 4 is even, so 1480 is divisible by 8. Here’s a little more about its factors:

  • 1480 is a composite number.
  • Prime factorization: 1480 = 2 × 2 × 2 × 5 × 37, which can be written 1480 = 2³ × 5 × 37
  • 1480 has at least one exponent greater than 1 in its prime factorization so √1480 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1480 = (√4)(√370) = 2√370
  • 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 1480 has exactly 16 factors.
  • The factors of 1480 are outlined with their factor pair partners in the graphic below.

Something else about the number 1480:

1480 is the hypotenuse of FOUR Pythagorean triples:
456-1408-1480 which is 8 times (57-176-185)
480-1400-1480 which is (12-35-37) times 40
832-1224-1480 which is 8 times (104-153-185)
888-1184-1480 which is (3-4-5) times 296

1471 and Level 3

Today’s Puzzle:

There is only one factor that will work with the clue in the top row of this level 3 puzzle. Find it, then work your way down row by row until you have found all the factors. There is only one solution.

Factors of 147

  • 1471 is a prime number.
  • Prime factorization: 1471 is prime.
  • 1471 has no exponents greater than 1 in its prime factorization, so √1471 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1471 has exactly 2 factors.
  • The factors of 1471 are outlined with their factor pair partners in the graphic below.

How do we know that 1471 is a prime number? If 1471 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1471. Since 1471 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1471 is a prime number.

Another Fact about the Number 1471:

1471 is the 21st centered heptagonal number because
(7(21²)-7(21)+2)/2 = 1471.

1434 and Level 3

If you know the common factors of 14 and 8, then you will have an excellent start to solve this puzzle. Once you place the factors of 18 and 8 in their proper places, just work down the puzzle row by row until all the factors from 1 to 10 are found.

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Now I’ll share some facts about the puzzle number, 1434:

  • 1434 is a composite number.
  • Prime factorization: 1434 = 2 × 3 × 239
  • 1434 has no exponents greater than 1 in its prime factorization, so √1434 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 1434 has exactly 8 factors.
  • The factors of 1434 are outlined with their factor pair partners in the graphic below.

1434 is 234 in BASE 26 because
2(26²) + 3(26¹) + 4(26º) = 1434

1421 Square-shaped Sweet

This square-shaped puzzle is a Level 3 because you can start with clue 44 and work down the puzzle cell by cell to find the solution. You won’t get into a rhythm with this one, but logic will still help you to find the factors of the clues in order from top to bottom. That’s pretty sweet!

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now here are some sweet facts about the puzzle number 1421:

  • 1421 is a composite number.
  • Prime factorization: 1421 = 7 × 7 × 29, which can be written 1421 = 7² × 29
  • 1421 has at least one exponent greater than 1 in its prime factorization so √1421 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1421 = (√49)(√29) = 7√29
  • 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 1421 has exactly 6 factors.
  • The factors of 1421 are outlined with their factor pair partners in the graphic below.

1421 is also the hypotenuse of a Pythagorean triple:
980-1029-1421 which is (20-21-29) times 49.

1412 and Level 3

If you know the greatest common factor of 56 and 48, then you have taken the first step in solving this puzzle. Once you put the factors of 56 and 48 in the appropriate cells, work down from the top of the puzzle to the bottom, cell by cell, until you have put all the numbers from 1 to 10 in both the first column and the top row.

Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Here are a few facts about the puzzle number, 1412:

  • 1412 is a composite number.
  • Prime factorization: 1412 = 2 × 2 × 353, which can be written 1412 = 2² × 353
  • 1412 has at least one exponent greater than 1 in its prime factorization so √1412 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1412 = (√4)(√353) = 2√353
  • 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 1412 has exactly 6 factors.
  • The factors of 1412 are outlined with their factor pair partners in the graphic below.

1412 is the sum of two squares:
34² + 16² = 1412

1412 is the hypotenuse of a Pythagorean triple:
900-1088-1412 calculated from 34² – 16², 2(34)(16), 34² + 16²

1380 A Different Way to Look at the Logic

This puzzle looks a lot like puzzle #1378. The design is the same, but the clues are not in the same places. I made this puzzle to demonstrate that there is often more than one way to logically find all the factors of a puzzle. If it were a level 4 puzzle, the clues could be anywhere on the puzzle. But since it is a level 3 puzzle, start with the factors of 14 and 8, and then write the factors of 63 in the appropriate places. Continue with the clues in order from top to bottom until all the factors have been found.

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some information about the number 1380:

  • 1380 is a composite number.
  • Prime factorization: 1380 = 2 × 2 × 3 × 5 × 23, which can be written 1380 = 2² × 3 × 5 × 23
  • 1380 has at least one exponent greater than 1 in its prime factorization so √1380 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1380 = (√4)(√345) = 2√345
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1380 has exactly 24 factors.
  • The factors of 1380 are outlined with their factor pair partners in the graphic below.

There are MANY possible factor trees for 1380. Here’s one of them:

1380 is the hypotenuse of a Pythagorean triple:
828-1104-1380 which is (3-4-5) times 276

1378 and Level 3

The first row with a clue has a 14 in it. Use that 14 and the clue in the same column, to figure out where to put one of the factor pairs of 14 in this puzzle.  Only use factor pairs where both numbers are from 1 to 10. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle. The completed puzzle will look like a multiplication table with the factors in a different order.

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here is some information about the puzzle number, 1378:

  • 1378 is a composite number.
  • Prime factorization: 1378 = 2 × 13 × 53
  • 1378 has no exponents greater than 1 in its prime factorization, so √1378 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 1378 has exactly 8 factors.
  • The factors of 1378 are outlined with their factor pair partners in the graphic below.

1378 is the 52nd triangular number because (52)(53)/2 = 1378.

1378 is the hypotenuse of FOUR Pythagorean triples:
222-1360-1378 which is 2 times (111-680-689)
530-1272-1378 which is (5-12-13) times 106
728-1170-1378 which is 26 times (28-45-53)
800-1122-1378 which is 2 times (400-561-689)