4.1. Question 1 Question 1 13 Marks 1.1 Calculate the multiplicative inverse of 9 modulo 23 . Show your calculations (2 Marks) 1.2 Find two numbers whose greatest common divisor is 12 and least common multiple is 360 . Provide a detailed explanation of the steps involved in determining these two numbers (2 Marks) 1.3 A tech company is managing two servers. Server A processes data every 18 minutes, while Server B processes data every 24 minutes. The IT team wants to schedule routine checks on both servers so that the checks happen simultaneously, and the downtime is minimized. They need to find the longest time interval (in minutes) after which both servers will require maintenance at the same time. a. What is the greatest common divisor of the processing intervals for Server A and Server B? (2 Marks) b. What is the least common multiple of the processing intervals for Server A and Server B? (2 Marks)

To calculate the multiplicative inverse of 9 modulo 23, we need to find an integer \(x\) such that \(9x \equiv 1 \mod 23\). Using the Extended Euclidean Algorithm, we find that \(9 \cdot 18 \equiv 162\), and when we take \(162 \mod 23\), we get \(1\). Thus, the multiplicative inverse of 9 modulo 23 is \(18\). To find two numbers with a GCD of 12 and an LCM of 360, we can use the relationship between GCD, LCM, and the product of two numbers: If \(a\) and \(b\) are the two numbers, then \(GCD(a, b) \times LCM(a, b) = a \times b\). So, \(12 \times 360 = a \times b\) gives \(a \times b = 4320\). Let’s say \(a = 12x\) and \(b = 12y\) with \(GCD(x, y) = 1\). Hence, \(LCM(a, b) = 12xy\), setting up the equation \(12xy = 360\) leads to \(xy = 30\). Thus, possible pairs \((x, y)\) could be \((5, 6)\) or \((6, 5)\), giving us numbers \(60\) and \(72\) as solutions. The greatest common divisor (GCD) of the processing intervals for Server A (18 minutes) and Server B (24 minutes) can be determined using the method of prime factorization. The prime factorizations are \(18 = 2 \cdot 3^2\) and \(24 = 2^3 \cdot 3\). The GCD is found by taking the lowest powers of all prime factors present, which results in \(GCD(18, 24) = 6\). For the least common multiple (LCM) of the processing intervals for Server A (18 minutes) and Server B (24 minutes), we also use the prime factorization method. The LCM is determined by taking the highest powers of all prime factors: \(2^3\) from 24 and \(3^2\) from 18. This leads us to \(LCM(18, 24) = 2^3 \cdot 3^2 = 72\). Thus, scheduling checks every 72 minutes ensures both servers will require maintenance at the same time.