What is the common factor needed for 8 and 24 to make this puzzle work? Is it 2, 4, or 8? How about for 12 and 36? Is it 3, 4, 6, or 12? Don’t guess which common factor to use. In each case, all but one of the choices will be eliminated by using logic. It won’t be easy, but if you are determined, you can solve this puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 1044-1053

Here are some facts about the number 1048:

• 1048 is a composite number.
• Prime factorization: 1048 = 2 × 2 × 2 × 131, which can be written 1048 = 2³ × 131
• The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1048 has exactly 8 factors.
• Factors of 1048: 1, 2, 4, 8, 131, 262, 524, 1048
• Factor pairs: 1048 = 1 × 1048, 2 × 524, 4 × 262, or 8 × 131
• Taking the factor pair with the largest square number factor, we get √1048 = (√4)(√262) = 2√262 ≈ 32.372828

1048 can be written as the difference of two squares two different ways:
263² – 261² = 1048
133² – 129² = 1048

1048 can also be expressed as 2 times a factor pair 3 different ways:
2(524)(1)
2(262)(2)
2(131)(4)

Those facts make 1048 a leg in these FIVE obscure Pythagorean triples:
1048-137286-137290 calculated from 263² – 261², 2(263)(261), 263² + 261²
1048-34314-34330 calculated from 133² – 129², 2(133)(129), 133² + 129²
1048-274575-274577 calculated from 2(524)(1), 524² – 1², 524² + 1²
1048-68640-68648 calculated from 2(262)(2), 262² – 2², 262² + 2²
1048-17145-17177 calculated from 2(131)(4), 131² – 4², 131² + 4²

This puzzle consists of six sets of three numbers. Find the common factor of each set of clues so that ALL of the factors involved are a number from 1 to 12, and you’ll solve this puzzle. Have fun!

Here are a few facts about the number 1045:

Obviously, 1045 can be evenly divided by 5, but since 1-0+4-5 = 0, we know that 1045 is also divisible by 11.

• 1045 is a composite number.
• Prime factorization: 1045 = 5 × 11 × 19
• 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 1045 has exactly 8 factors.
• Factors of 1045: 1, 5, 11, 19, 55, 95, 209, 1045
• Factor pairs: 1045 = 1 × 1045, 5 × 209, 11 × 95, or 19 × 55
• 1045 has no square factors that allow its square root to be simplified. √1045 ≈ 32.32646

1045 is the hypotenuse of a Pythagorean triple:
627-836-1045 which is (3-4-5) times 209.

1045 is also palindrome 171 in BASE 29 because 1(29²) + 7(29) + 1(1) = 1045

All of the clues in this puzzle have three common factors, but only one of those three factors won’t put a number greater than twelve in either the first column or the top row. Can you figure out what that common factor is as well as all the other factors that belong in this puzzle?

Now I’ll share some information about the number 1044:

• 1044 is a composite number.
• Prime factorization: 1044 = 2 × 2 × 3 × 3 × 29, which can be written 1044 = 2² × 3² × 29
• The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1044 has exactly 18 factors.
• Factors of 1044: 1, 2, 3, 4, 6, 9, 12, 18, 29, 36, 58, 87, 116, 174, 261, 348, 522, 1044
• Factor pairs: 1044 = 1 × 1044, 2 × 522, 3 × 348, 4 × 261, 6 × 174, 9 × 116, 12 × 87, 18 × 58 or 29 × 36
• Taking the factor pair with the largest square number factor, we get √1044 = (√36)(√29) = 6√29 ≈ 32.31099

30² + 12² =1044

1044 is the hypotenuse of a Pythagorean triple:
720-756-1044 calculated from 2(30)(12), 30² – 12², 30² + 12².
It is also (20-21-29) times 36.

1044 is the sum of twin primes: 521 + 523 = 1044

1044 looks interesting a few other bases:
It’s 414 in BASE 16 because 4(16²) + 1(16) + 4(1) = 1044,
TT in BASE 35 (T is 29 base 10) because 29(35) + 29(1) = 29(35 + 1) = 29(36) = 1044, and T0 in BASE 36 because 29(36) = 1044

I’ve already published the two level 5 puzzles that are in this week’s set of puzzles. If going from a level 4 puzzle to a level 6 puzzle is too big of a jump for you, then try either one of those two level 5 puzzles first. You can find them as well as this puzzle in the link below the puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

Here are a few facts about the number 1042:

• 1042 is a composite number.
• Prime factorization: 1042 = 2 × 521
• 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 1042 has exactly 4 factors.
• Factors of 1042: 1, 2, 521, 1042
• Factor pairs: 1042 = 1 × 1042 or 2 × 521
• 1042 has no square factors that allow its square root to be simplified. √1042 ≈ 32.28002

31²  + 9² = 1042

1042 is the hypotenuse of a Pythagorean triple:
558-880-1042 calculated from 2(31)(9), 31²  – 9², 31²  + 9²

1042 is also a palindrome in a couple of bases:
It’s 868 in BASE 11 because 8(121) + 6(11) + 8(1) = 1042, and
2C2 in BASE 20 (C is 12 in base 10) because 2(400) + 10(20) + 2(1) = 1042

See clues 63 and 72 near the top of this puzzle? Start there and work down cell by cell to find all the factors that will make this puzzle become a multiplication table. Only write each number from 1 to 10 once in the top row and once in the first column.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

Now here are some facts about the number 1040:

• 1040 is a composite number.
• Prime factorization: 1040 = 2 × 2 × 2 × 2 × 5 × 13, which can be written 1040 = 2⁴ × 5 × 13
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1040 has exactly 20 factors.
• Factors of 1040: 1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 40, 52, 65, 80, 104, 130, 208, 260, 520, 1040
• Factor pairs: 1040 = 1 × 1040, 2 × 520, 4 × 260, 5 × 208, 8 × 130, 10 × 104, 13 × 80, 16 × 65, 20 × 52 or 26 × 40
• Taking the factor pair with the largest square number factor, we get √1040 = (√13)(√65) = 4√65 ≈ 32.24903.

1040 is the sum of the twelve prime numbers from 61 to 109.
It is also the sum of these four prime numbers:
251 + 257 + 263 + 269 = 1040

1040 is the hypotenuse of FOUR different Pythagorean triples:
256-1008-1040 which is 16 times (16-63-65)
400-960-1040 which is (5-12-13) times 80
528-896-1040 which is 16 times (33-56-65)
624-832-1040 which is (3-4-5) times 208

I like the way 1040 looks in a couple of other bases:
It’s 2020 in BASE 8 because 2(8³) + 2(8) = 2(520) = 1040, and
it’s palindrome 3A3 in BASE 17 (A is 10 base 10) because 3(17²) + 10(17) + 3(1) = 1040