Hey guys, here are some problems that I've been having a bit of trouble with. Hopefully you guys can help out a bit. Thank you so much!
1. Find the smallest positive integer a for which 3≡amod22.
2. Prove or disprove: a^3≡amod3, for all integers a
3. Prove or disprove: a^4=amod4
4. A check scheme we saw in class, used in some countries for passport numbers, is also used to produce six
digit identification numbers that are placed on the ears of cattle. A number a1a2…a6 is valid when
a1+3a2 +7a3 +a4 +3a5 +7a6 ≡0mod10. Find B so that 3129B7 is a valid number.
5. Determine all x in which [(4x + 3)/2] (lower floor function)=5
6. Determine the positive integers g for which g=gcd (2n+3, 4n−8) for some integer n.
7. Prove that a base 10 integer dn dn−1…d1d0 is divisible by 4 if and only if d1d0 is divisible by 4. For example,
7,927,356 is divisible by 4, since 56 is divisible by 4.
8. Find by argument, not brute force, the largest positive integer a for which a2 divides 9! .