K-10. If a and b are integers, 30, 310,5 ta, and 56, prove that a =...
Prove or give a counterexample: For any integers b and c and any positive integer m, if b ≡ c (mod m) then b + m ≡ c (mod m).
6. Prove that if a and b are odd integers, then a2 is divisible by 8. 7. Prove that if a is an odd integer, then ta + (a + 2)?+ (a +4)2 +1) is divisible by 12.
Let A, B be an n × n matrices. Prove that [10%] ABt = B tA t
(c) contrapositive positiv 2. (a) Prove that for all integers n and k where n >k>0, (+1) = 0)+2). (b) Let k be a positive integer. Prove by induction on n that ¿ () = 1) for all integers n > k. 3. An urn contains five white balls numbered from 1 to 5. five red balls numbered from 1 to 5 and fiv
13. Prove that for all integers b, if b is odd then b is odd 13. Prove that for all integers b, if b is odd then b is odd
8. Given integers m and 1<a<m, with am, prove that the equation ar = 1 (mod m) has no solution. (This means that here is no 1 appearing in the multiplication table mod m, in front of any of the divisors of m. That is, if m is composite, and a is a factor of m then a has no multiplicative inverse in mod m.)
Ulscrete Mathematics a. Prove that k (*)=n (1 - 1) for integers n and k with 15ks n, using a i. combinatorial proof: (3 marks) ii. algebraic proof. (3 marks)
P.4 Prove that for any set of integers {ao, aj, a2,..., ax), the integer n=ax. 10% +ax-1·10k-1 + ... + 01.10+ 0 is congruent to E-01–1)' a; (mod 11). What significance does this hold when the ai are restricted to the set {0,1,2,3,4,5,6,7,8,9}?
6.2 Prove that for fixed positive integers k and n, the number of partitions of n is equal to the number of partitions of 2n + k into n + k parts.
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...