MAST232 - How to do 2D Graphs in Mathematics - Mathematics Assignment Help

Assignment Task

Question 1
A natural number n with even number of digits is called heavy if the number formed with the first half of the number is the quarter of the number formed with the second half of digits. For example, for the number 1040 is heavy, since 10 is the quarter of 40. How many positive natural numbers, between 105 and 106 - 1, are heavy?

Question 2
Let f(n) = n2 - 79 n + 1601 and g (n) = 11 n2 + 55 n + 43.
1. How many of the numbers f(0), f(1), ..., f(100) are prime? Similarly, how many of g(0), g(1), ..., g(100) are prime?
2. Consider F and G the sets of numbers generated by f and g, respectively, in point a). Are F and G equals as sets?
3. List the set of elements of F and G that are not primes.

Question 3
The lcm(a,b) represents the least common multiple of integers a and b. For a given number n, we can calculate the average of the lcm of n and each number less or equal to it. That is, we can calculate the average of lcm(n,k) with 1 ≤ k ≤ n.
Example: Let’s consider n = 8. Then: lcm(8,1)=8; lcm(8,2)=8; lcm(8,3)=24; lcm(8,4)=8; lcm(8,5)=40; lcm(8,6)=24; lcm(8,7)=56. The average of these numbers is 168
8 = 21. So, for n=8, we associate the number 21.
Your task: identify the largest difference for the average of lcm() between two consecutive numbers, for numbers from 1 to 10000. For which two numbers does this occur for the first time?
Example. For n=9, we have the average lcm equal to 31. Then, the difference between the average lcm of 8 and 9 is 10.
Observation. Problem partially based on Euler Project problem 448.

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