1414 Your Math Education Post Will Add So Much to This Month’s Carnival!

Have you written a blog post that would bring delight to a preschool, K-12 or homeschool mathematics teacher or student? Then submit it to this month’s Playful Math Education Blog Carnival or message me on Twitter by Friday, September 20th! I’m hosting the carnival this month, and I would love to read your post. So come join the fun!

Today’s puzzle looks a little like a wild, but fun? carnival ride. The numbers 36 and 12 went together on the ride. They managed to stay with each other but the ride went so fast, you can see 36 and 12 in two different places at the same time. There’s also poor number 40. You can see it in THREE places at the same time.

Oh my! Can you use logic to find where the numbers 1 to 10 need to go in both the first column and the top row so that this wild ride will behave like a multiplication table? It’s a level 5 so it won’t be easy to find its unique solution. Are you brave enough to try?

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

That puzzle’s number is 1414. Let me tell you a little about that number:

  • 1414 is a composite number.
  • Prime factorization: 1414 = 2 × 7 × 101
  • 1414 has no exponents greater than 1 in its prime factorization, so √1414 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 1414 has exactly 8 factors.
  • The factors of 1414 are outlined with their factor pair partners in the graphic below.

1414 is also the hypotenuse of a Pythagorean triple:
280-1386-1414 which is 14 times (20-99-101)

 

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1398 and Level 5

You might find this puzzle to be a little tricky, but if you always use logic before you write any of the factors, you should succeed!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here is some information about the number 1398:

  • 1398 is a composite number.
  • Prime factorization: 1398 = 2 × 3 × 233
  • 1398 has no exponents greater than 1 in its prime factorization, so √1398 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 1398 has exactly 8 factors.
  • The factors of 1398 are outlined with their factor pair partners in the graphic below.

1398 is the hypotenuse of a Pythagorean triple:
630-1248-1398 which is 6 times (105-208-233)

1383 and Level 5

Level 5 puzzles aren’t any harder than level 4 puzzles unless I trick you into starting with the common factor of a pair of clues that have more than one possibility. You won’t let me trick you, will you?

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

Here are a few facts about the number 1383:

  • 1383 is a composite number.
  • Prime factorization: 1383 = 3 × 461
  • 1383 has no exponents greater than 1 in its prime factorization, so √1383 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 1383 has exactly 4 factors.
  • The factors of 1383 are outlined with their factor pair partners in the graphic below.

1383 is the hypotenuse of a Pythagorean triple:
783-1140-1383 which is 3 times (261-380-461)

1361 and Level 5

If you carefully use logic instead of guessing and checking you can find the unique solution to this puzzle without tearing your hair out!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1361:

  • 1361 is a prime number.
  • Prime factorization: 1361 is prime.
  • 1361 has no exponents greater than 1 in its prime factorization, so √1361 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1361 has exactly 2 factors.
  • The factors of 1361 are outlined with their factor pair partners in the graphic below.

How do we know that 1361 is a prime number? If 1361 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1361. Since 1361 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1361 is a prime number.

1361 is the first prime number after 1327. That was 34 numbers ago!

1361 is the sum of two squares:
31² + 20² = 1361

1361 is the hypotenuse of a Pythagorean triple:
561-1240-1361 calculated from 31² – 20², 2(31)(20), 31² + 20²

Here’s another way we know that 1361 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 20² = 1361 with 31 and 20 having no common prime factors, 1361 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1361. Since 1361 is not divisible by 5, 13, 17, or 29, we know that 1361 is a prime number.

1354 Solving a Level 5 Puzzle

What are the common factors of 16 and 4? Don’t guess which one to use. Use logic to figure it out as you find all the factors for this puzzle!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you get stuck, you can watch this video:

Now I’ll share some information about the puzzle number, 1354:

  • 1354 is a composite number.
  • Prime factorization: 1354 = 2 × 677
  • 1354 has no exponents greater than 1 in its prime factorization, so √1354 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 1354 has exactly 4 factors.
  • The factors of 1354 are outlined with their factor pair partners in the graphic below.

1354 is the sum of two squares:
27² + 25² = 1354

1354 is the hypotenuse of a Pythagorean triple:
104-1350-1354 which is 2 times (52-675-677)
and can also be calculated from 27² – 25², 2(27)(25), 27² + 25²

1338 and Level 5

Finding the most logical place to start a level 5 puzzle is only a little bit trickier than for a level 4 puzzle. Will you figure it out and not be tricked?

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the number 1338:

  • 1338 is a composite number.
  • Prime factorization: 1338 = 2 × 3 × 223
  • 1338 has no exponents greater than 1 in its prime factorization, so √1338 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 1338 has exactly 8 factors.
  • The factors of 1338 are outlined with their factor pairs in the graphic below.

1338 is in a couple of Pythagorean triples:
1338-447560-447562 calculated from 2(669)(1), 669² – 1², 669² + 1² and
1338-49720-49738 calculated from 2(223)(3), 223² – 3², 223² + 3²

1328 Christmas Tree

 

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

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

Now I’ll share some facts about the puzzle number, 1328:

  • 1328 is a composite number.
  • Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
  • Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
  • Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
  • Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

 

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree:

1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328

1316 Hard Candy

 

When I have a cold or a cough, I often have a piece of hard candy in my mouth. These red cinnamon candies are tasty, but they aren’t very good for soothing throats! Will this red hot cinnamon candy puzzle be too hard for you to solve? It may be a little bit of a challenge, but I’m sure you can solve it if you let logic be your guide the entire time. 

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

Now I’ll share some information about the number 1316:

  • 1316 is a composite number.
  • Prime factorization: 1316 = 2 × 2 × 7 × 47, which can be written 1316 = 2² × 7 × 47
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1316 has exactly 12 factors.
  • Factors of 1316: 1, 2, 4, 7, 14, 28, 47, 94, 188, 329, 658, 1316
  • Factor pairs: 1316 = 1 × 1316, 2 × 658, 4 × 329, 7 × 188, 14 × 94, or 28 × 47
  • Taking the factor pair with the largest square number factor, we get √1316 = (√4)(√329) = 2√329 ≈ 36.27671

1316 is the difference of two squares two different ways:
330² – 328² = 1316
54² – 40² = 1316 

1307 and Level 5

This puzzle shows 10 of the 100 products in a 10 × 10 multiplication table. Can you figure out where to put the factors? There’s only one way that works!

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Here are some facts about the number 1307:

  • 1307 is a prime number.
  • Prime factorization: 1307 is prime.
  • The exponent of prime number 1307 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1307 has exactly 2 factors.
  • Factors of 1307: 1, 1307
  • Factor pairs: 1307 = 1 × 1307
  • 1307 has no square factors that allow its square root to be simplified. √1307 ≈ 36.15245

How do we know that 1307 is a prime number? If 1307 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1307 ≈ 36.2. Since 1307 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1307 is a prime number.

1307 is palindrome 797 in BASE 13 because 7(13²) + 9(13) + 7(1) = 1307

1293 and Level 5

Only factors from 1 to 12 are allowed in the factor pairs of the clues in these puzzles. Find a row or column with only one allowable common factor to start this puzzle. As you progress, factors for other rows or columns will be eliminated. Keep at it, and you will succeed!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll share a few facts about the number 1293:

  • 1293 is a composite number.
  • Prime factorization: 1293 = 3 × 431
  • 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 1293 has exactly 4 factors.
  • Factors of 1293: 1, 3, 431, 1293
  • Factor pairs: 1293 = 1 × 1293 or 3 × 431
  • 1293 has no square factors that allow its square root to be simplified. √1293 ≈ 35.95831

1293 is palindrone 141 in BASE 34 because 34² + 4(34) + 1 = 1293