## Question

1. The remainder when 5¹⁴⁰ is divided by 11. [15 points]

2. All positive integers n for which 11 ≡ 4 (mod n). [10 points]

3. The general solution to 15x ≡ 11 (mod 73). [10 points

4. All integers between 1000 and 2000 that leave a remainder of 2 when divided by

5, a remainder of 4 when divided by 7 and a remainder of 1 when divided by 11. [15 points]

II. Word Problems

5. An all-you-can-eat restaurant charges 750 pesos for adults and 650 pesos for senior citizens. Children eat for free. At the end of the day, the restaurant had collected 25,000 pesos. What is the largest number of people who could have eaten there that day? What is the smallest number of people? [15 points]

III. Proving

6. Let a, b € Z such that (a,b) = 1. Prove that (a + b, a² - ab + b²) = 1 or 3. [10 points]

7. Let Nn be the integer whose decimal expansion consists of n consecutive ones.

For example, N2 = 11 and N4 = 1, 111 . Show that (Nn,Nmm) = N(n,m)- [10 points]

8. A number N is divisible by 7, 11 and 13 if and only if the alternating sum and difference of blocks of three ddigits is divisible by 7, 11 and 13. [15 points]

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