1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding.

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

Now I’ll share a few facts about the puzzle number, 1413:

  • 1413 is a composite number.
  • Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
  • 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
  • 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 1413 has exactly 6 factors.
  • The factors of 1413 are outlined with their factor pair partners in the graphic below.

1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

Advertisements

1397 and Level 4

I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do!

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

Now I’ll write a little bit about the puzzle number, 1397:

  • 1397 is a composite number.
  • Prime factorization: 1397 = 11 × 127
  • 1397 has no exponents greater than 1 in its prime factorization, so √1397 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 1397 has exactly 4 factors.
  • The factors of 1397 are outlined with their factor pair partners in the graphic below.

1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397

1382 and Level 4

After solving a couple of level 3 puzzles, you are ready to give a level 4 puzzle a try. It isn’t any more difficult to solve than a level 3, except that the clues are not given in a logical order. Don’t let that stop you from succeeding!

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

Now I’ll share some information about the number 1382:

  • 1382 is a composite number.
  • Prime factorization: 1382 = 2 × 691
  • 1382 has no exponents greater than 1 in its prime factorization, so √1382 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 1382 has exactly 4 factors.
  • The factors of 1382 are outlined with their factor pair partners in the graphic below.

1382 is in one Pythagorean triple:
1382-477480-477482 calculated from 2(691)(1), 691² – 1², 691² + 1²

1360 and Level 4

There is only one way to write the numbers 1 to 12 in the top row and the first column of this puzzle so that the puzzle could turn into a multiplication table. Can you find the solution?

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

Here are some facts about the number 1360:

  • 1360 is a composite number.
  • Prime factorization: 1360 = 2 × 2 × 2 × 2 × 5 × 17, which can be written 1360 = 2⁴ × 5 × 17
  • 1360 has at least one exponent greater than 1 in its prime factorization so √1360 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1360 = (√16)(√85) = 4√85
  • The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1360 has exactly 20 factors.
  • The factors of 1360 are outlined with their factor pair partners in the graphic below.

1360 is the sum of two squares in two different ways:
36² + 8² = 1360
28² + 24² = 1360

1360 is the hypotenuse of FOUR Pythagorean triples:
208-1344-1360 which is 16 times (13-84-85)
576-1232-1360 which is 16 times (36-77-85)
640-1200-1360 which is (8-15-17) times 80
816-1088-1360 which is (3-4-5) times 272

1353 How to Solve a Level 4 Puzzle

Try solving this level 4 puzzle. If you need help with it, I explain the steps in the video below the puzzle:

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

Now I’ll tell you a little bit about the puzzle number, 1353:

  • 1353 is a composite number.
  • Prime factorization: 1353 = 3 × 11 × 41
  • 1353 has no exponents greater than 1 in its prime factorization, so √1353 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 1353 has exactly 8 factors.
  • The factors of 1353 are outlined with their factor pair partners in the graphic below.

1353 is the hypotenuse of a Pythagorean triple:
297-1320-1353 which is 33 times (9-40-41)

1337 and Level 4

If you can solve a level 3 puzzle, give this one a try. You will have to locate the best clues to use in the beginning and then what is the most logical clue to use next over and over, but I think you can do that!

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

Here is some information about the puzzle number, 1337:

  • 1337 is a composite number.
  • Prime factorization: 1337 = 7 × 191
  • 1337 has no exponents greater than 1 in its prime factorization, so √1337 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 1337 has exactly 4 factors.
  • The factors of 1337 are outlined with their factor pairs in the graphic below.

1337 is the difference of two squares two ways:
99² – 92² = 1337
669² – 668² = 1337

1325 Hockey Stick

 

If someone you know loves hockey and wants a fun way to practice multiplication facts, this hockey stick could be the perfect gift.

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

  • 1325 is a composite number.
  • Prime factorization: 1325 = 5 × 5 × 53, which can be written 1325 = 5² × 53
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1325 has exactly 6 factors.
  • Factors of 1325: 1, 5, 25, 53, 265, 1325
  • Factor pairs: 1325 = 1 × 1325, 5 × 265, or 25 × 53
  • Taking the factor pair with the largest square number factor, we get √1325 = (√25)(√53) = 5√53 ≈ 36.40055

1325 is the sum of two squares in three different ways:
29² + 22² = 1325 
34² + 13² = 1325
35² + 10² = 1325

1325 is the hypotenuse of SEVEN Pythagorean triples:
115-1320-1325 which is 5 times (23-264-265)
357-1276-1325 calculated from 29² – 22², 2(29)(22), 29² + 22²
371-1272-1325 which is (7-24-25) times 53
480-1235-1325 which is 5 times (96-247-265)
700-1125-1325 calculated from 2(35)(10), 35² – 10², 35² + 10²
795-1060-1325 which is (3-4-5) times 265
884-987-1325 calculated from 2(34)(13), 34² – 13², 34² + 13²

1315 Peppermint Stick

 

A red, green, and white peppermint stick can be used to stir hot chocolate or enjoyed as a candy for a long time. Will it take you very long to solve this peppermint stick puzzle? May you find sweet success as you find all the factors!

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

Now I’ll share some facts about the number 1315:

  • 1315 is a composite number.
  • Prime factorization: 1315 = 5 × 263
  • 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 1315 has exactly 4 factors.
  • Factors of 1315: 1, 5, 263, 1315
  • Factor pairs: 1315 = 1 × 1315 or 5 × 263
  • 1315 has no square factors that allow its square root to be simplified. √1315 ≈ 36.26293

1315 is the sum of three consecutive prime numbers:
433 + 439 + 443 = 1315

1315 is the hypotenuse of a Pythagorean triple:
789-1052-1315 which is (3-4-5) times 263

1305 and Level 4

This level 4 puzzle has clues taken from a standard multiplication table, but the factors of those clues are not in their usual places. Can you figure out where the numbers 1 to 10 belong in this multiplication table?

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

Now I’ll share some information about the puzzle’s number, 1305:

  • 1305 is a composite number.
  • Prime factorization: 1305 = 3 × 3 × 5 × 29, which can be written 2905 = 3² × 5 × 29
  • 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 1305 has exactly 12 factors.
  • Factors of 1305: 1, 3, 5, 9, 15, 29, 45, 87, 145, 261, 435, 1305
  • Factor pairs: 1305 = 1 × 1305, 3 × 435, 5 × 261, 9 × 145, 15 × 87, or 29 × 45
  • Taking the factor pair with the largest square number factor, we get √1305 = (√9)(√145) = 3√145 ≈ 36.12478.

1305 is the sum of two squares in two ways:
36² + 3² = 1305
27² + 24² = 1305

153-1296-1305 calculated by 27² – 24², 2(27)(24), 27² + 24²
216-1287-1305 calculated by 2(36)(3), 36² – 3², 36² + 3²
783-1044-1305 which is (3-4-5) times 261
900-945-1305 which is (20-21-29) times 45

1292 and Level 4

Finding the Factors in this puzzle will require you to use logic and you will get a bonus: your knowledge of the multiplication table will be a little better than it was before.

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

Now I’ll give you some information about today’s puzzle number:

  • 1292 is a composite number.
  • Prime factorization: 1292 = 2 × 2 × 17 × 19, which can be written 1292 = 2² × 17 × 19
  • 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 1292 has exactly 12 factors.
  • Factors of 1292: 1, 2, 4, 17, 19, 34, 38, 68, 76, 323, 646, 1292
  • Factor pairs: 1292 = 1 × 1292, 2 × 646, 4 × 323, 17 × 76, 19 × 68, or 34 × 38
  • Taking the factor pair with the largest square number factor, we get √1292 = (√4)(√323) = 2√323 ≈ 36.9444

1292 is the hypotenuse of a Pythagorean triple:
608-1140-1292 which is (8-15-17) times 76