Number theory and discrete mathematics – university Year 1 maths test to download as PDF

Use the Euclidean algorithm to find \(\gcd(390,72)\) and write it as \(390x+72y\).

1. Solve \(9x\equiv6\pmod{15}\). (2 marks)
2. Find \(2^{50}\bmod7\). (2 marks)

1. How many arrangements are there of the letters of STATISTICS? (2 marks)
2. A committee of 5 is chosen from 7 men and 6 women. How many committees contain at least 2 women? (2 marks)

A graph has 7 vertices with degrees \(1,2,2,3,3,3,4\).

1. How many edges does it have? (1 mark)
2. Can it be a tree? Explain. (1 mark)
3. Does it have an Euler trail? Explain. (2 marks)

A sequence of savings tokens satisfies \(a_n=4a_{n-1}-4a_{n-2}\) with \(a_0=1\), \(a_1=4\).

1. Compute \(a_2\) and \(a_3\). (1 mark)
2. Solve the recurrence and check your formula against \(a_3\). (3 marks)