1 Perfect Square

1 has 1 factor. 2 has 2 factors…that is the end of that pattern because no number greater than 2 equals its number of factors.

  • 1 is not a prime number, and 1 is not a composite number. 1 is in a category all by itself. It is classified as a unit.
  • 1 has no Prime factorization.
  • p⁰ = 1, where p is any prime number, so 1 is a factor of every prime number and every composite number.
  • 1 is also the only number to have exactly 1 factor.
  • Factors of 1: 1
  • Factor pairs: 1 = 1 x 1
  • √1 = 1. Since its square root is a whole number, 1 is a perfect square.

1(n) = n and n ÷ 1 = n for every number n.

Also 1⁰ = 1, 1¹ = 1, 1² = 1, 1³ = 1, 1⁴ = 1, 1⁵ = 1, 1⁶ = 1, 1⁷ = 1, 1⁸ = 1, 1⁹ = 1. In fact, 1 raised to any power equals 1. Even 1⁻⁹⁸⁷⁶⁵⁴³²¹⁰ = 1.

Not only that, but any number (EXCEPT 0) raised to the zeroth power is equal to 1.

One of my college professors wrote something like the following on the board to show why 0º is NOT defined:

When 1 is a clue in the FIND THE FACTORS puzzle, write 1 in both the corresponding factor row and the corresponding factor column.


Like most people, you probably know how to fill in a multiplication table even if it looks like this:

standard random table

The numbers that are given on a table can be called clues. The table above has 20 clues. What is the least number of clues that a table could have and still only have one way to fill it out?


Although the table above has just nine clues, there is still only one way to complete it. Nine is the fewest number of clues that will still yield a unique solution.  All of those clues would have to be perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. One of the clues will be missing, but it isn’t difficult to figure out where the missing clue should go.  Always find the factors on the outside of the puzzle BEFORE writing down the products on the inside of the puzzle.

The puzzle above is rated difficulty level ONE because you only need to know 10 multiplication facts to find all the factors. If this puzzle is too easy for you, you can try a more difficult puzzle. Levels FOUR, FIVE, or SIX will be much more challenging, even for adults.

This link, 10 Factors 2013-10-28, will bring up an excel file with the puzzles that are on this post.  After you enable editing, you can print the puzzles or type the factor answers directly onto the excel file.

An answer key will be posted one week after a puzzle is published.

If you don’t want to open the excel file, the rest of the puzzles will be printed below. If you cut and paste them on a document, you can make them any size you want.




If you want to check your work, the answers are given in a tab of the excel file that was published a week later: 10 factors 2013-11-04.