What is the greatest common factor of 28 and 63? If you know, then you can probably figure out this puzzle!

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

Here are some facts about the number 1283:

• 1283 is a prime number.
• Prime factorization: 1283 is prime.
• The exponent of prime number 1283 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1283 has exactly 2 factors.
• Factors of 1283: 1, 1283
• Factor pairs: 1283 = 1 × 1283
• 1283 has no square factors that allow its square root to be simplified. √1283 ≈ 35.81899

How do we know that 1283 is a prime number? If 1283 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1283 ≈ 35.8. Since 1283 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1283 is a prime number.

1283 is the sum of the seventeen prime numbers from 41 to 109,
AND it is the sum of the thirteen primes from 71 to 131.

Can you find the factors from 1 to 10 that make the twelve clues in the puzzle the correct products for this scrambled multiplication table?

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

Here is some information about the number 1282:

• 1282 is a composite number.
• Prime factorization: 1282 = 2 × 641
• 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 1282 has exactly 4 factors.
• Factors of 1282: 1, 2, 641, 1282
• Factor pairs: 1282 = 1 × 1282 or 2 × 641
• 1282 has no square factors that allow its square root to be simplified. √1282 ≈ 35.80503

1282 is the sum of two squares:
29² +  21² = 1282

1282 is the hypotenuse of a primitive Pythagorean triple:
400-1218-1282 calculated from 29² –  21², 2(29)( 21), 29² +  21²

The 21, 29, and 400 above are related to another Pythagorean triple:
20-21-29 because 20² = 400, 21² = 441 and 29² = 841. Thus,
400 + 441 = 841. Pretty cool!

Can you write the numbers from 1 to 10 in the top row and the first column so that the given clues will make this puzzle work like a multiplication table? That’s how you solve the puzzle!

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

Now I’ll write a little bit about the number 1281:

• 1281 is a composite number.
• Prime factorization: 1281 = 3 × 7 × 61
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1281 has exactly 8 factors.
• Factors of 1281: 1, 3, 7, 21, 61, 183, 427, 1281
• Factor pairs: 1281 = 1 × 1281, 3 × 427, 7 × 183, or 21 × 61
• 1281 has no square factors that allow its square root to be simplified. √1281 ≈ 35.79106

1281 is also the sum of consecutive prime numbers in two different ways:
167 + 173 + 179 + 181 + 191 + 193 + 197 = 1281
241 + 251 + 257 + 263 + 269 = 1281

1281 is the hypotenuse of a Pythagorean triple:
231-1260-1281 which is 21 times (11-60-61)

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.

Try your hand at solving this level 4 puzzle. Your ability to do so might just surprise you!

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

Since this is my 1278th post, I’ll write a little bit about that number:

• 1278 is a composite number.
• Prime factorization: 1278 = 2 × 3 × 3 × 71, which can be written 1278 = 2 × 3² × 71
• 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 1278 has exactly 12 factors.
• Factors of 1278: 1, 2, 3, 6, 9, 18, 71, 142, 213, 426, 639, 1278
• Factor pairs: 1278 = 1 × 1278, 2 × 639, 3 × 426, 6 × 213, 9 × 142, or 18 × 71,
• Taking the factor pair with the largest square number factor, we get √1278 = (√9)(√142) = 3√142 ≈ 35.74913

1278 and the four numbers immediately preceding it are the smallest consecutive numbers for which 4 of the 5 numbers each have exactly 12 factors.

1278 is the sum of eight consecutive prime numbers:
139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 = 1278

What’s the only common factor or 12 and 11? Write factors of 12 and 11 where they belong on the puzzle below. Then starting back up at the top of the puzzle, go down the puzzle writing the appropriate factors cell by cell until you’re done. You can do this!

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

Here’s some information about the number 1273:

• 1273 is a composite number.
• Prime factorization: 1273 = 19 × 67
• 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 1273 has exactly 4 factors.
• Factors of 1273: 1, 19, 67, 1273
• Factor pairs: 1273 = 1 × 1273 or 19 × 67
• 1273 has no square factors that allow its square root to be simplified. √1273 ≈ 35.67913

1273 is also palindrome 10011111001 in BASE 2.

Some of the factor pairs needed to solve this puzzle may be easier for you to find than others, but I’m sure you can still find all of them. Give it a try!

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

Now I’ll write a little bit about the number 1272:

• 1272 is a composite number.
• Prime factorization: 1272 = 2 × 2 × 2 × 3 × 53, which can be written 1272 = 2³ × 3 × 53
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1272 has exactly 16 factors.
• Factors of 1272: 1, 2, 3, 4, 6, 8, 12, 24, 53, 106, 159, 212, 318, 424, 636, 1272
• Factor pairs: 1272 = 1 × 1272, 2 × 636, 3 × 424, 4 × 318, 6 × 212, 8 × 159, 12 × 106, or 24 × 53
• Taking the factor pair with the largest square number factor, we get √1272 = (√4)(√318) = 2√318 ≈ 35.66511

1272 is the sum of four consecutive prime numbers, and it is the sum of two consecutive prime numbers:
311 + 313 + 317 + 331 = 1272
631 + 641 = 1272

1272 is the hypotenuse of a Pythagorean triple:
672-1080-1272 which is 24 times (28-45-52)

All the clues in this level 1 puzzle have a greatest common factor. If you can figure out what that GCF is, then you’ve completed the first step necessary to solve the puzzle.

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

Now I’ll write a few facts about the number 1271:

• 1271 is a composite number.
• Prime factorization: 1271 = 31 × 41
• 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 1271 has exactly 4 factors.
• Factors of 1271: 1, 31, 41, 1271
• Factor pairs: 1271 = 1 × 1271 or 31 × 41
• 1271 has no square factors that allow its square root to be simplified. √1271 ≈ 35.65109

1271 is the sum of the nineteen prime numbers from 29 to 107.
It is also the sum of three consecutive primes: 419 + 421 + 431 = 1271

1271 is the hypotenuse of a Pythagorean triple:
279-1240-1271 which is 31 times (9-40-41)