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By VICTOR SHOUP

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The most function of those lectures is first to in short survey the basic con­ nection among the illustration conception of the symmetric staff Sn and the idea of symmetric services and moment to teach how combinatorial equipment that come up certainly within the concept of symmetric features result in effective algorithms to specific a variety of prod­ ucts of representations of Sn by way of sums of irreducible representations.

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3. Let a, b, n, n ∈ Z with n > 0, n > 0, and gcd(n, n ) = 1. Show that if a ≡ b (mod n) and a ≡ b (mod n ), then a ≡ b (mod nn ). 4. Let a, b, n ∈ Z such that n > 0 and a ≡ b (mod n). Show that gcd(a, n) = gcd(b, n). 5. Prove that for any prime p and integer x, if x2 ≡ 1 (mod p) then x ≡ 1 (mod p) or x ≡ −1 (mod p). 6. Let a be a positive integer whose base-10 representation is a = (ak−1 · · · a1 a0 )10 . Let b be the sum of the decimal digits of a; that is, let b := a0 + a1 + · · · + ak−1 . Show that a ≡ b (mod 9).

Moreover, show that if α ∈ Z∗n , then these identities hold for all integers k1 , k2 . 4 Euler’s phi function Euler’s phi function φ(n) is defined for positive integer n as the number of elements of Z∗n . Equivalently, φ(n) is equal to the number of integers between 0 and n − 1 that are relatively prime to n. For example, φ(1) = 1, φ(2) = 1, φ(3) = 2, and φ(4) = 2. 11. For any positive integer n, we have φ(d) = n, d|n where the sum is over all positive divisors d of n. Proof. Consider the list of n rational numbers 0/n, 1/n, .

K with j = i. That is to say, for i, j = 1, . . , k, we have wi ≡ δij (mod nj ), where δij := 1 if i = j, 0 if i = j. Now define k z := wi ai . i=1 One then sees that k z≡ k wi ai ≡ i=1 δij ai ≡ aj (mod nj ) for j = 1, . . , k. i=1 Therefore, this z solves the given system of congruences. Moreover, if z ≡ z (mod n), then since ni | n for i = 1, . . , k, we see that z ≡ z ≡ ai (mod ni ) for i = 1, . . , k, and so z also solves the system of congruences. Finally, if z solves the system of congruences, then z ≡ z (mod ni ) for i = 1, .

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