In the earlier post, students were asked to determine the length of the expansion of 300!, the number’s last digit, and the number of that digit at the end. My students had powerful TI-Nspires available, so I deliberately asked about a number that would be difficult to deal with by hand, encouraging them to let computation tools do accurate work while they did the thinking.
Because 10 is one of the factors of 300!, the last digit was rapidly declared to be 0. The two length questions took longer.
Most computed 300! on their Nspires and were amazed by the machines’ exact answers for such a large number even though they have difficulty displaying the result. The following image is from Wolfram Alpha. It shows the entire number, but also some additional information I wanted my students to reason for themselves. Unfortunately, the Wolfram Alpha result also states the length of the number, so I was happier that they were “limited” to their Nspires.
Following is a summary of the track of their reasoning.
- One student unintentionally pressed ctrl-enter when computing 300! and got , not the exact answers his classmates discovered. After some thought about the length of numbers and their scientific notation equivalents, he declared the number to be 615 digits long.
- Another student declared the same answer after computing . The class knew this answer was correct, but didn’t initially understand why until one suggested that , convincing them that both the scientific notation and the logarithm were giving the same results for this problem, but in different forms.
Then attention turned to the number of zeros, a more challenging question if judged by how long it took them to decide on an answer.
- A couple students opted for a careful count, arriving at 74 zeros at the end of 300!. But most were unwilling to do that count on their Nspires because of the challenge of the display–a fact we well understood when we created the problem.
- Most of the others realized quickly that the problem hinges on the number of 10s used to create 300!. Some used this to get a first guess of 30 zeros because 300! includes factors of 10, 20, 30, …, 290, 300. This was upgraded to 33 to account for the double factors of 10 in 100, 200, and 300.
- Another spoke up to say that factors of 5, 15, 25, etc. could create additional final 0s when multiplied by other factors containing additional 2s. Because these factors where spaced half way between the earlier multiples of 10, the class estimate jumped another 30 to 63.
- Many realized that this needed to be increased further because every multiple of 25 contained two 5s. There are 12 of those (25, 50, …, 300) increasing their count to 75. The numbers 125 & 250 added two more to the count because each had three 5s in their factorizations giving a final guess of 77.
- By this point, one questioned why we were going through all this trouble to account for the locations of both the 5s and 10s. Because every 10 contained a factor of 5, this really just depended on the number of 5s. Ultimately, she said, it also didn’t matter where the 5s happened. We just needed to know how many 5s there were in the factorization of 300!. Her Nspire had a factor command, so she guessed correctly that it could handle the following which suggested the correct answer of 74 zeros. I asked if she was worried about having enough 2s, and she noted that there were 296 of those, so like limiting factors in her chemistry class last year, the zeros at the end of 300! were limited by the number of 5s.
- It took some guidance, but they eventually discovered that the difference between their initial 77 and their final 74 could be accounted for by the unintended over-counting of 100, 200, and 300 in their shift from counting 10s to counting 5s.
A secondary realization from this great conversation was that the factorization of a factorial number created a list of all of the prime numbers less than or equal to the integer in the factorial if you ignored all the exponents. This Wolfram Alpha page shows this result nicely.