Last week I was in a high school algebra classroom and the teacher was helping students get back into the swing of the school year and doing mathematics. He was playing a game with the students in which he would display a number using a random number generator and the students would have to use all of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as well as operations and grouping symbols to equal the number displayed on the screen.
For example, if the random number generator produced the number 17 a winning answer could be
17 = (4)²+1+((3+5+6+7+8+9)0) = 16 + 1 + 0 = 17
The first team to figure out a way to represent the number got 5 points. If another team could represent the number in a different way they earned 2 points. If a team checked another answer and found an error, they earned a point also.
The game was exciting! It was loud! People were yelling about the order of operations and exponents! They were challenging each other about math! It was music to my ears!
Then, the random number generator displayed the number 147. It started off like any other round . . . whispering, writing, stopping, writing some more . . . and then something interesting happened–no one could figure out a way to combine the 10 digits to get the number 147.
I left that day thinking about the number 147 . . . was it impossible to generate the number using all 10 digits? If so, how many other numbers could not be generated using this method also? As I was driving back to my office I began thinking about the properties of 147. First, I wondered is 147 prime? But I quickly figured out that is was not prime (I used the test for divisibility by 3–1+4+7=12, which is divisible by 3, so 147 is divisible by 3). The prime factors of 147 were 7 and 3, in fact 147 = 7²(3).
Now I had a strategy! Was there a way I could combine 10 digits to equal 7, 7, and 3? If so I could represent 147 in the way that was so tricky! Guess what? I could! And now its your turn! Show me how you represented 147! Can you do this in a way that doesn’t build off of the prime factorization of the number? Good Luck!