Integer Factorization
Let \(N = pq\) be a positive integer which is the product of two large prime numbers \(p\) and \(q\). Let \(R\) be the remainder when \(p+q\) is divided by \(10^9+7\). Hint: the difference between \(p\) and \(q\) is less than \(10^{80}\) (the current estimate for the number of atoms in the observable universe).
| Part A | Find \(R\) when \(N = \) 37834082197. | |
|---|---|---|
| Part B | Find \(R\) when \(N = \) 31259182824182977579361. | |
| Part C | Find \(R\) when \(N = \) 279680497756621349605906986937174040586 145180581133159093315713686803955381832 199906939110100292778501757276492336647 733009077451211345797330743124737928346 315459217030828079849109738375831203116 482566885088773675647315143800526347195 967243145966681580005600894548499860991 115107257526176229258827758033769663. |