For this note I assume that you know the basics properties of divisibility including those of the congruence relation . See for example Modular arithmetics
I will show some nice problems involving the congruence relation, most of them are very useful in problem solving, thus we can consider this problems as lemmas.
Problem 1. If then there exists an integer such that . is called the multiplicative inverse of modulo .
Solution. Consider the numbers . If for , then , since we conclude that and then . Thus the numbers are pairwise distinct modulo , since this set consists of numbers then it is a complete residue system modulo . In particular, for some we have .
Problem 2. Let be an integer:
If is prime then . [Wilson's Theorem]
If is composite and then .
Hint for the first part. If is an odd prime, prove that the set can be divided in pairs where and are inverse each other.
Problem 3. If is a prime and then .
Solution. We know that then . The number is divisible by while is not, then is divisible by .
Problem 4. If is a prime then .
Hint. Use the previous problem.
Now, you can practice with the following problems. To avoid confusion I will call these problems N1, N2, etc.
Problem N1. ¿Does there exist a positive integer for wich the set can be partitioned in two subsets and such that the product of the elements of is equal to the product of the elements of ?
Problem N2. Let be a prime. For each denote by the remainder when is divided by , thus . Calculate .
Please post your comments and solutions!
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Problem N1:
We claim the answer is no.Lemma: No element in the set can be a multiple of 19.
Proof: Assume that an element in the set is divisible by 19. Then, since the set is composed of 18 consecutive integers, there can be no other element in the set that is divisible by 19. However, then the product of the elements in the two sets cannot be equal, as one has a factor of 19, while the other doesn't, a contradiction. □
Since the set is composed of 18 consecutive integers, none of which is divisible by 19, the set must be equivalent to the set {1,2,⋯,18} when reduced modulo 19. If this set can be partitioned into two subsets with the same product, the product of the elements must be a square. By Wilson's theorem, the product of the elements is congruent to −1 modulo 19. However, it is well-known that −1 is a quadratic residue modulo an odd prime p if and only if that prime is 1 modulo 4. Since 19≡3(mod4), we conclude that no such integer n exists. ■
Problem N2:
We claim the answer is 2p2(p−1).
Lemma: ip+(p−i)p≡0(modp2), and ri+rp−i=p2.
Proof: By the binomial theorem, ip+(p−i)p≡ip+pp−(1p)pp−1i+⋯−(p−2p)p2ip−2+(p−1p)pip−1−ip(modp2)≡ip+0−0+⋯−0+(p−1p)pip−1−ip(modp2)≡ip+p2ip−1−ip(modp2)≡0(modp2) However, also note that, for 1≤i≤p−1, p2∤ip. Since 0<ri<p2, 0<rp−i<p2, and ri+rp−i≡0(modp2), we must have that ri+rp−i=p2, as desired. □
By the lemma, and since p−1 is even, the set of integers r1,r2,⋯,rp−1 can be paired, ri with rp−i, such that the sum of each pair is p2. The total number of pairs is 2p−1, and so we conclude r1+r2+⋯+rp−1=2p2(p−1). ■
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Awesome! Really nice job
Excellent!!
Since Problem 4 has no proof, I will write one, though the hint really gives it away..
By the binomial theorem we have that, (a+b)p=k=0∑p(kp)akbp−k
By Problem 3, we know that 1≤k<p⇒(kp)≡0(modp), and so all of the summands except for the first and last ones are zeroed. We are then left with,
(a+b)p≡(0p)ap+(pp)bp≡ap+bp(modp) □