David W. answered • 08/13/15
Experienced Prof
Problem Statement: Sam counted the lines of a page in his book. Counting by threes gave a remainder of 2; counting by fives also gave a remainder of 2; and counting by sevens gave a remainder of 5.
Problem Description: On a single page, there are N (that Sam counted) lines.
Question: What is the value of N (the number of lines on the specific page)?
Answer: (PLZ wait on the answer, I want to write the solution first)
First, having made a poster of "The Sieve of Eratosthenes" for a project in eighth grade, I recognize that 2, 3, and 5 are the first prime numbers. I hope you are studying prime numbers, because I will use them in this solution.
On my poster, I made rows of length 30 (2*3*5) and drew lines through columns deleting everything below 2, all of the column for 4,6,8,10,12,14,16,18,20,22,24,26,28,30 and everything below 3 and all of the column for 6,9,12,15,18,21,24,27,30 and everything below 5 and all of the column for 10,15,20,25,30. This left a lot fewer numbers to consider as possible primes when I began dividing by 7, 11, 13, etc.
What's that have to do with this problem? Well, the N that we seek has a remainder of 2 when divided by each of 3 and 5, and a remainder of 5 when divided by 7. There are a lot of such numbers, but we want to know which ones make sense for what Sam counted (also, it is interesting when the difference of two numbers is a prime number).
After a few more years, I began to program computers and learned about "exhaustive enumeration." That's the process of considering all possible values to see which ones fit the criteria. The problem criteria is "remainder of 2 when divided by 3 AND remainder of 2 when divided by 5 AND remainder of 5 when divided by 7." All that is needed is to test each number to see whether this is true. So, I asked my computer to do this (it is very, very fast and very, very accurate) and it started to list them: 47, 152, 257, 362, 467, 572, 677, 782, 887, 992, ... I assume you can divide by 2, 3, and 7 to double-check that all these numbers (and many, many more) fit the criteria.
The only thing to do now is to determine what size page Sam could have been reading (and how high Sam can count). For me, that leaves multiple possibilities (I read e-books). You will have to determine your answer.
Proof narrative: There is not a "proof" here because there is not a single answer. The remainders of numbers when divided by prime numbers occur continuously. When those remainders are also prime, we have other properties (for later math) that are very interesting.
