Math 232-061
Extra Problems on Number Theory
(16 Problems)
1. Prove or disprove: if a | bc , then a | b or a | c .
2. Prove that a positive integer is square if and only if the exponent of each of its
prime divisors is even.
3. Is the number 111
...
11 prime or composite.



8
4.
5.
6.
7.
8.
9.
Prove that gcd( n, n  1)  1 or 2 for every integer n .
Find all positive integers m  1 that satisfy 70  2 mod m .
Let n be integer. What values can n 2 assume modulo 7.
Let a and b be odd integers. Show that a 2  b 2  2 mod 4 .
Find the least residue of 2 50 modulo 13.
Find the least residue of 25111 modulo 7.
100
10. Find the least residue of
 k! modulo 5.
k 1
11. Let n be a positive integer. What is the least residue of 11  14 n  1 modulo 5.
12. If x  2 mod 7 , then find the value of 2 x 3  x 2  3x  10 modulo 7.
13. What are the possible values of the last digit (units digit) of n 2 , where n is
integer.
14. Prove that every integer is congruent modulo 9 to the sum of its digits. (For
example, 21  2  1mod 9 .)
15. Prove that every prime greater than 3 is congruent to 1 or 5 modulo 6.
16. Use Mathematical induction to prove 5 n  4n  1 mod 16 .
Good luck,
Dr. Ibrahim Al-Rasasi