1308 and Level 6

Can you find the factors that will turn this puzzle into a multiplication table? It’s a level 6 so you might find it to be a challenge. Use logic. Don’t guess and check.

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

Here are a few facts about the puzzle number, 1308:

  • 1308 is a composite number.
  • Prime factorization: 1308 = 2 × 2 × 3 × 109, which can be written 1308 = 2² × 3 × 109
  • 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 1308 has exactly 12 factors.
  • Factors of 1308: 1, 2, 3, 4, 6, 12, 109, 218, 327, 436, 654, 1308
  • Factor pairs: 1308 = 1 × 1308, 2 × 654, 3 × 436, 4 × 327, 6 × 218, or 12 × 109
  • Taking the factor pair with the largest square number factor, we get √1308 = (√4)(√327) = 2√327 ≈ 36.16628

1308 is the hypotenuse of a Pythagorean triple:
720-1092-1308 which is 12 times (60-91-109)

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. Thank you Stetson.edu for alerting me to that fact.

 

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1294 and Level 6

You will need to use multiplication facts from 1 to 12, logic, and perseverance to solve this tricky level 6 puzzle. Good luck!

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

Here are some facts about the number 1294:

  • 1294 is a composite number.
  • Prime factorization: 1294 = 2 × 647
  • 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 1294 has exactly 4 factors.
  • Factors of 1294: 1, 2, 647, 1294
  • Factor pairs: 1294 = 1 × 1294 or 2 × 647
  • 1294 has no square factors that allow its square root to be simplified. √1294 ≈ 35.97221

1294 is also the sum of the twenty prime numbers from 23 to 107.

1286 and Level 6

This level 6 puzzle can be solved by using logic and basic knowledge of the multiplication table. Stay focused, and you will get it done!

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

Here are a few facts about the number 1286:

  • 1286 is a composite number.
  • Prime factorization: 1286 = 2 × 643
  • 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 1286 has exactly 4 factors.
  • Factors of 1286: 1, 2, 643, 1286
  • Factor pairs: 1286 = 1 × 1286 or 2 × 643
  • 1286 has no square factors that allow its square root to be simplified. √1286 ≈ 35.86084

1286 is also the sum of six consecutive prime numbers:
197 + 199 + 211 + 223 + 227 + 229 = 1286

1280 and Level 6

To me, today’s level 6 puzzle looks a little like a puppy dog. If you know or use a multiplication table, then with proper training, finding the factors of this puzzle will be no problem.

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

I’d like to tell you a little about the number 1280:

  • 1280 is a composite number.
  • Prime factorization: 1280 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5, which can be written 1280 = 2⁸ × 5
  • The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1280 has exactly 18 factors.
  • Factors of 1280: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 640, 1280
  • Factor pairs: 1280 = 1 × 1280, 2 × 640, 4 × 320, 5 × 256, 8 × 160, 10 × 128, 16 × 80, 20 × 64, or 32 × 40
  • Taking the factor pair with the largest square number factor, we get √1280 = (√256)(√5) = 16√5 ≈ 35.77709.

1280 is the sum of the fourteen prime numbers from 61 to 127. Do you know what those prime numbers are?

32² + 16² = 1280

1280 is the hypotenuse of a Pythagorean triple:
768-1024-1280 which is (3-4-5) times 256
That triple can also be calculated from 32² – 16², 2(32)(16), 32² + 16²

Since 1280 is the 5th multiple of 256, I would expect that a number close to 1280 would be the 500th number whose square root could be simplified. That number was 1275, just five numbers ago.

1257 and Level 6

Both 6 and 12 are allowable common factors of 60 and 12. Likewise, both 8 and 12 are allowable common factors of 96 and 72. In each case, only one of those common factors will work with this puzzle. Don’t guess and check each one. Study the other clues and at least one wrong common factor will be eliminated. Have fun solving it!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll write a few things about the number 1257:

  • 1257 is a composite number.
  • Prime factorization: 1257 = 3 × 419
  • 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 1257 has exactly 4 factors.
  • Factors of 1257: 1, 3, 419, 1257
  • Factor pairs: 1257 = 1 × 1257 or 3 × 419
  • 1257 has no square factors that allow its square root to be simplified. √1257 ≈ 35.4542

1257 is the difference of two squares two different ways:
211² – 208² = 1257
629² – 628² = 1257

1257 is palindrome 393 in BASE 19

1250 and Level 6

The clues in one of the columns for this puzzle as well as one of the rows are 9 and 3. You will need to figure out where to put the factors 1, 3, 3, and 9 to make those clues work. You might think it doesn’t matter where you write those factors, but believe me, it does matter. My advice: Don’t start with those clues. Find another logical place to start. Good luck with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Here are some facts about the number 1250:

  • 1250 is a composite number.
  • Prime factorization: 1250 = 2 × 5 × 5 × 5 × 5, which can be written 1250 = 2 × 5⁴
  • The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(4 + 1) = 2 × 5 = 10. Therefore 1250 has exactly 10 factors.
  • Factors of 1250: 1, 2, 5, 10, 25, 50, 125, 250, 625, 1250
  • Factor pairs: 1250 = 1 × 1250, 2 × 625, 5 × 250, 10 × 125, or 25 × 50
  • Taking the factor pair with the largest square number factor, we get √1250 = (√625)(√2) = 25√2 ≈ 35.35534

1250 is the sum of consecutive squares two different ways:
193 + 197 + 199 + 211 + 223 + 227 = 1250
619 + 631 = 1250

1250 is the sum of two squares THREE different ways:
31² + 17² = 1250
25² + 25² = 1250
35² + 5² = 1250

1250 is the hypotenuse of FOUR Pythagorean triples:
750-1000-1250 which is (3-4-5) times 250,
672-1054-1250 which is 2 times (336-527-625) and
can also be calculated from 31² – 17², 2(31)(17), 31² + 17²,
440-1170-1250 which is 10 times (44-117-125), and
350-1200-1250 which is (7-24-25) times 50 and
can also be calculated from 2(35)(5), 35² – 5², 35² + 5²

1238 and Level 6

If you use logic, you can figure out the solution to this puzzle. You will have to study all the clues just to know where to start, but I think you’ll find a lot of satisfaction in finding the solution.

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Now I’ll share some facts about the number 1238:

  • 1238 is a composite number.
  • Prime factorization: 1238 = 2 × 619
  • 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 1238 has exactly 4 factors.
  • Factors of 1238: 1, 2, 619, 1238
  • Factor pairs: 1238 = 1 × 1238 or 2 × 619
  • 1238 has no square factors that allow its square root to be simplified. √1238 ≈ 35.18522

Because of its prime factors, I know that 1238 is part of only one Pythagorean triple:
1238-383160-383162

1238 is a palindrome in two other bases:
It’s 646 in BASE 14, and
it’s 383 in BASE 19.

1229 and Level 6

The only common factors permitted for 32 and 40 in this puzzle are 4 and 8, but which one will work for this puzzle? Likewise, you must decide if 3 or 6 is the right common factor for 18 and 30. Don’t guess which factor to use. Study the other clues and let logic guide your decisions until the unique solution is found. Have fun with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

This is my 1229th post, so I will tell you a little bit about the number 1229:

  • 1229 is a prime number.
  • Prime factorization: 1229 is prime.
  • The exponent of prime number 1229 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1229 has exactly 2 factors.
  • Factors of 1229: 1, 1229
  • Factor pairs: 1229 = 1 × 1229
  • 1229 has no square factors that allow its square root to be simplified. √1229 ≈ 35.057

How do we know that 1229 is a prime number? If 1229 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1229 ≈ 35.1. Since 1229 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1229 is a prime number.

1229 is the sum of three consecutive prime numbers:
401 + 409 + 419 = 1229

1229 is the sum of two square numbers:
35² + 2²  = 1229

1229 is the hypotenuse of a primitive Pythagorean triple:
140-1221-1229 calculated from 2(35)(2), 35² – 2², 35² + 2²

Here’s another way we know that 1229 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 2² = 1229 with 35 and 2 having no common prime factors, 1229 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1229 ≈ 35.1. Since 1229 is not divisible by 5, 13, 17, or 29, we know that 1229 is a prime number.

 

1216 and Level 6

Is 6 or 9 the needed common factor for 54 and 36?
The other one will be the common factor for 72 and 18.

Will 5 or 10 be the common factor of 30 and 50?
Will 2 or 4 be the common factor of 16 and 4?

Don’t guess the answers to those questions! The other clues in the puzzle will help you find the answers. Will you be stumped or will you triumph?

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’ll share some information about the number 1216:

  • 1216 is a composite number.
  • Prime factorization: 1216 = 2 × 2 × 2 × 2 × 2 × 2 × 19, which can be written 1216 = 2⁶ × 19
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1216 has exactly 14 factors.
  • Factors of 1216: 1, 2, 4, 8, 16, 19, 32, 38, 64, 76, 152, 304, 608, 1216
  • Factor pairs: 1216 = 1 × 1216, 2 × 608, 4 × 304, 8 × 152, 16 × 76, 19 × 64, or 32 × 38
  • Taking the factor pair with the largest square number factor, we get √1216 = (√64)(√19) = 8√19 ≈ 34.87119

1216 is the sum of the ten prime numbers between 100 and 150:
101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1216

1216 is the 19th nonagonal number because
19(7·19 – 5)/2 = 1216, in other words because 19 × 64 = 1216

1206 and Level 6

If you carefully study all the clues in this Level 6 puzzle and use logic, you should be able to solve the puzzle. Stick with it and you’ll succeed!

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are some facts about the number 1206:

  • 1206 is a composite number.
  • Prime factorization: 1206 = 2 × 3 × 3 × 67, which can be written 1206 = 2 × 3² × 67
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1206 has exactly 12 factors.
  • Factors of 1206: 1, 2, 3, 6, 9, 18, 67, 134, 201, 402, 603, 1206
  • Factor pairs: 1206 = 1 × 1206, 2 × 603, 3 × 402, 6 × 201, 9 × 134, or 18 × 67,
  • Taking the factor pair with the largest square number factor, we get √1206 = (√9)(√134) = 3√134 ≈ 34.72751

Notice that 6·201 = 1206. Not very many numbers can equal themselves by using their own digits in a different way with +, -, ×, ÷, and/or parenthesis. That fact makes 1206 only the seventeenth Friedman Number.

1206 is also the sum of the twenty prime numbers from 19 to 103.