# Lagrange's Theorem Problems

## Question

Lagrange's Theorem: classification of finite abelian groups.

1. (a) Prove that the product of integers relatively prime to n is relatively prime to n, and moreover, the following cancellation law holds: if c is relatively prime to n and ac=bc mod n, then a=b mod n.
(b) Derive from this that the set Zā* of remainders mod n relatively prime to n forms a group with respect to multiplication modulo n. Following Euler, the order of this group is denoted phi (n) (the Greek letter "phi")
(c) Use Lagrange's Theorem to prove the following Euler's generalization of Fermat's Little Theorem: If x is relatively prime to n, then x raised to the power phi(n) has remainder 1 modulo n.
2. Find the remainder of 300^300 (300 to the power 300) upon division by 1001.
3. Show that every abelian group of order 1001 is cyclic.
4. Classify up to isomorphism all abelian groups of order 16. Find the place of Z*āā (multiplicative group of odd remainders modulo 32) in your classification.
5. Classify up to isomorphism all abelian groups of order 360.

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