Practice problems for the final
I'm having some writer's block coming up with extra problems, but I'll keep working at it. Here are three so far. (As usual, it's trickier than what's actually on the final.)
1a. Let a,b be natural numbers, a < b. Assume there exist x,y in the naturals such that xa + yb = gcd(a,b). Show y < 10.
b. Give an example where x > 10.
2. What are the last two decimal digits of 112008?
3a. Draw an example of finite sets X,Y,Z and functions f:X->Y, g:Y->Z where |X| > |Y|, f is not onto, but the composite g o f is onto.
b. Prove that |Y| is not equal to |Z| (not just in your example, but in any example).
4. In each of the following: can you turn the following two congruences into one, and if so, what?
v1. b congruent to 2 mod 6, b congruent to 7 mod 10.
v2. b congruent to 1 mod 6, b congruent to 7 mod 10.
5. Say b == a+7 mod 27, and a2 == b2 mod 27.
Figure out a,b mod 27.
1a. Let a,b be natural numbers, a < b. Assume there exist x,y in the naturals such that xa + yb = gcd(a,b). Show y < 10.
b. Give an example where x > 10.
2. What are the last two decimal digits of 112008?
3a. Draw an example of finite sets X,Y,Z and functions f:X->Y, g:Y->Z where |X| > |Y|, f is not onto, but the composite g o f is onto.
b. Prove that |Y| is not equal to |Z| (not just in your example, but in any example).
4. In each of the following: can you turn the following two congruences into one, and if so, what?
v1. b congruent to 2 mod 6, b congruent to 7 mod 10.
v2. b congruent to 1 mod 6, b congruent to 7 mod 10.
5. Say b == a+7 mod 27, and a2 == b2 mod 27.
Figure out a,b mod 27.
posted by Allen Knutson @ 8:57 PM
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