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HOMEWORK II MATH-UA 0248-001 THEORY OF NUMBERS

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HOMEWORK II

MATH-UA 0248-001 THEORY OF NUMBERS due on September, 22, 2017

1. Prove that the product of any three consecutive integers is divisible by 6.

2. Establish that if a is an odd integer, then24|a(a2−1) (Hint: prove that the square of an odd integer is of the form8k+ 1).

3. Use the Euclidean algorithm to compute gcd(54321,9876).

4. Find one integer solution of the equation 62x+ 34y= 2.

5. Find all integer solutions of the equation 15x+ 33y= 7.

6. Find all integer solutions of the equation 19x+ 99y= 3.

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