Today’s puzzle comes gift-wrapped just for you. Figuring out the solution to this puzzle is about as easy as ripping gift-wrap off a present, too. What is the common factor in both parts of the ribbon? There is only one answer to that question that will not put any numbers greater than 12 where the factors go. So put the factors of the clues in the first column and top row so that this puzzle becomes a multiplication table (but with the factors in a different order than usual).

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here are some facts about the number 978:

It is made from 3 consecutive numbers, 7-8-9, so it is divisible by 3.

OEIS.org reminds us that it is the sum of four consecutive fourth powers:
2⁴ + 3⁴ + 4⁴ + 5⁴ = 978

It is the sum of two consecutive prime numbers:
487 + 491 = 978

I like the way it looks when written in a couple of other bases:
Palindrome 696 in BASE 12 because 6(12²) + 9(12) + 6(1) = 978
369 in BASE 17 because 3(17²) + 6(17) + 9(1) = 978

• 978 is a composite number.
• Prime factorization: 978 = 2 × 3 × 163
• 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 978 has exactly 8 factors.
• Factors of 978: 1, 2, 3, 6, 163, 326, 489, 978
• Factor pairs: 978 = 1 × 978, 2 × 489, 3 × 326, or 6 × 163
• 978 has no square factors that allow its square root to be simplified. √978 ≈ 31.27299

This puzzle has eleven clues, some of which are designed to trick you possibly. Will you be tricked, or will you use logic to figure out where to put the factors from 1 to 10 in the first column and the top row? The finished puzzle looks like a multiplication table but with the factors out of numerical order.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Here are a few facts about the number 975:

975 is the hypotenuse of SEVEN Pythagorean triples. Some factor pairs of 975 are highlighted in red next to those triples.
108-969-975 which is 3 times (36-323-325)
240-945-975 which is 15 times (16-63-65)
273-936-975 which is (7-24-25) times 39
375-900-975 which is (5-12-13) times 75
495-840-975 which is 15 times (33-56-65)
585-780-975 which is (3-4-5) times 195
612-759-975 which is 3 times (204-253-325)

1(5⁴) + 2(5³) + 3(5²) + 4(5¹) + 5(5⁰) = 975

You might think that last fact means that 975 is 12345 in base 5, but it isn’t. The only digits used in base 5 are 0, 1, 2, 3, and 4.

Here is 975 written in some different bases:
1111001111 in BASE 2 because 2⁹ + 2⁸ + 2⁷+ 2⁶ + 2³ + 2² + 2¹ + 2⁰ = 975
33033 in BASE 4 because 3(4⁴) + 3(4³) + 0(4²) + 3(4¹) + 3(4⁰) = 3(256 + 64 + 4 + 1) = 3(325) = 975
1717 in BASE 8 because 1(8³) + 7(8²) + 1(8¹) + 7(8⁰) = 975
303 in BASE 18 because 3(18²) + 0(18¹) + 3(18⁰) = 3(324 + 1) = 3(325) = 975

• 975 is a composite number.
• Prime factorization: 975 = 3 × 5 × 5 × 13, which can be written 975 = 3 × 5² × 13
• 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 975 has exactly 12 factors.
• Factors of 975: 1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975
• Factor pairs: 975 = 1 × 975, 3 × 325, 5 × 195, 13 × 75, 15 × 65, or 25 × 39,
• Taking the factor pair with the largest square number factor, we get √975 = (√25)(√39) = 5√39 ≈ 31.22499

This puzzle has ten clues and, like always, it has only one solution. If you can figure out where to put the factors 1 to 10 in the first column as well as the top row, then you will have found that solution.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

This is my 974th post.

974 is the sum of three consecutive square numbers:
17² + 18² + 19² = 974

• 974 is a composite number.
• Prime factorization: 974 = 2 × 487
• 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 974 has exactly 4 factors.
• Factors of 974: 1, 2, 487, 974
• Factor pairs: 974 = 1 × 974 or 2 × 487
• 974 has no square factors that allow its square root to be simplified. √974 ≈ 31.20897

If this were a Find the Factors 1 – 12 puzzle, then clues 72 and 24 would have three different possible common factors, but that isn’t the kind of puzzle it is. It is just a Find the Factors 1 – 10 puzzle, and there is just one possible common factor that will put only factors from 1 – 10 in the first column and the top row. Do you know that common factor? If you do, you will be well on your way to solving the puzzle!

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Here are a few facts about the number 973:

It is palindrome 191 in BASE 27 because 1(27²) + 9(27) + 1(1) = 973

• 973 is a composite number.
• Prime factorization: 973 = 7 × 139
• 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 973 has exactly 4 factors.
• Factors of 973: 1, 7, 139, 973
• Factor pairs: 973 = 1 × 973 or 7 × 139
• 973 has no square factors that allow its square root to be simplified. √973 ≈ 31.1929