Ch 03 Test 2022-10a

Based on 2020-2021 CCML 1 Freshmen contest.

1. (opint) Given integers m and n, find the number of ordered pairs of natural numbers $(a,b)$ with $1\le a \le m$ and $1 \le b \le n$, so that $a/b$ is a natural number.

2. (countdiv) Count the number of divisors of a given number.

3. Write a function to determine whether or not a number is prime.

    isPrime :: Int -> Boolean
    isPrime n = False -- FIXME
    check_prime = [ isPrime 7,
                    not (isPrime 9),
                    isPrime 137,
                    not (isPrime 91),
                    isPrime 1231,
                    not (isPrime 58483)]
   

4. (lpd) Determine the largest prime divisor of a number. If asked to find the largest prime divisor of 1 or 0, your function should give a message with error.

    lpd :: Int -> Int
    lpd n = 0
    check_lpd = [7 == lpd 210,
                 251 == lpd 58483 ]
   

5. (fromDigits) Take a list of digits and produce a base ten number. Example: the list [1,6,5] should produce the number 165.

6. How many four digit numbers consisting only of digits 1, 2, or 3 are divisible by 3? Make a list of them.

7. Lockers are numbered consecutively beginning with one. If the digit three is used exactly 114 times, determine the largest locker number that could be used.

   Example: using exactly five threes, you can count to 31, so locker 5 == 31. The list with the 3’s in bold is: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31.

   Another example: locker 8 == 33 because the next two numbers (32 and 33) contain three more 3s.

   Advanced: expand your work for problem 7 into a function that takes in the number of threes you are looking for (like 114) and puts out the largest locker number that can be used.