### 978 A Gift-Wrapped Puzzle

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.

Stetson.edu 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

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