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407HW1

The document contains homework problems for MA 407, focusing on topics such as the existence of solutions to equations involving gcd, the uniqueness in the Fundamental Theorem of Arithmetic, properties of gcd with products, induction proofs regarding sums of cubes of consecutive integers, and the number of subsets in a set. Each problem requires mathematical proof or demonstration. The problems are designed to test understanding of number theory and combinatorics.

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quinn.gebeaux
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20 views1 page

407HW1

The document contains homework problems for MA 407, focusing on topics such as the existence of solutions to equations involving gcd, the uniqueness in the Fundamental Theorem of Arithmetic, properties of gcd with products, induction proofs regarding sums of cubes of consecutive integers, and the number of subsets in a set. Each problem requires mathematical proof or demonstration. The problems are designed to test understanding of number theory and combinatorics.

Uploaded by

quinn.gebeaux
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MA 407 HW 1

1. Let n and a be positive integers, and d = gcd(n, a). Show that the equation
ax mod n = 1 has a solution if and only if d = 1.
2. Prove the uniqueness portion of the Fundamental Theorem of Arithmetic
(for a precise statement, see Theorem 0.3. The existence potion is done
by induction in Example 13 of chapter 0).

3. Show that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a, c) = 1.


4. Use induction (instead of the modular arithmetic trick we did in class)
to prove that the sum of the cubes any three consecutive positive integers
is divisible by 9.
5. Prove that every set with n elements has exactly 2n subsets.

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