# 3.14 Computer Problems

1. Evaluate 

1. Find integers  and  with 

2. Find integers  and  with 

2. You are told that exactly one of the numbers



is prime and you have one minute to figure out which one. They do not have any prime factors less than  You may use modular exponentiation, but you may not use commands of the form “IsPrime[n]” or “NextPrime[n].” (This makes explicit Exercise 30 above.)

3. Find the last five digits of  (Note: Don’t ask the computer to print  It is too large!)

4. Look at the decimal expansion of  Find the consecutive digits 71, the consecutive digits 271, and the consecutive digits 4523 form primes. Find the first set of five consecutive digits that form a prime ( does not count as a five-digit number).

5. Solve 

6. Find all solutions to 

7. Find an integer such that when it is divided by 101 the remainder is 17, when it is divided by 201 the remainder is 18, and when it is divided by 301 the remainder is 19.

8. Let  Show that  Find an exponent  such that 

9. Let  Find  and  with  but 

10. Let 

1. Find the inverse of 

2. For which primes  does  not have an inverse mod ?

11. Find the square roots of 26055 mod the prime 34807.

12. Find all square roots of 1522756 mod 2325781.

13. Try to find a square root of 48382 mod the prime 83987, using the method of Section 3.9. Square your answer to see if it is correct. What number did you find the square root of?