Modular Arithmetic
... not prime. Let us assume that n + 1 is divisible by 2 and that n > 2. Then n + 1 is not a prime number. Now n + 2 is not divisible by 2. However, we could assume that n + 2 is divisible by 3 and n + 2 > 3. Then n + 2 is certainly not a prime either. Similarly we could assume that n + 3 is divisible ...
... not prime. Let us assume that n + 1 is divisible by 2 and that n > 2. Then n + 1 is not a prime number. Now n + 2 is not divisible by 2. However, we could assume that n + 2 is divisible by 3 and n + 2 > 3. Then n + 2 is certainly not a prime either. Similarly we could assume that n + 3 is divisible ...
Answers to some typical exercises
... remainder 0 when divided by 6, then we have done. If none of them have remainder 0, then there are at most 5 cases (pigeonhole) of the remainder. Thus, at least two of them must have the same remainder. The positive difference of these two is a subsequence whose sum is divisible by 6. ...
... remainder 0 when divided by 6, then we have done. If none of them have remainder 0, then there are at most 5 cases (pigeonhole) of the remainder. Thus, at least two of them must have the same remainder. The positive difference of these two is a subsequence whose sum is divisible by 6. ...
Chapter 8.10 - MIT OpenCourseWare
... since 1, 5, 7, and 11 are the only numbers in Œ0::12/ that are relatively prime to 12. More generally, if p is prime, then .p/ D p 1 since every positive number in Œ0::p/ is relatively prime to p. When n is composite, however, the function gets a little complicated. We’ll get back to it in the next ...
... since 1, 5, 7, and 11 are the only numbers in Œ0::12/ that are relatively prime to 12. More generally, if p is prime, then .p/ D p 1 since every positive number in Œ0::p/ is relatively prime to p. When n is composite, however, the function gets a little complicated. We’ll get back to it in the next ...