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Using the addition function over the natural numbers, give a recursive definition of multiplication of natural

numbers.

Question The integer square root of a positive integer $n$ is the largest integer whose square is less than or

equal to n $.$ For instance, the integer square roots of $15$ and $16$ are $3$ and $4,$ respectively. Given a primitive

recursive definition of this function.

Question Give a recursive definition of a function to find the highest common factor of two positive

integers.

Question Give a definition of the function below

orderTriple : : $($Integer$,$ Integer, Integer$)->($Integer$,$ Integer, Integer$)$

which puts the elements of a triple of three integers into ascending order.

Question Define a function to give the length of the perimeter of a geometrical shape, of type Shape. What

is the type of this function?

Question Add an extra constructor to Shape for triangles, and extend the functions isRound, area and

perimeter to include triangles.

Question Give a definition of a function below, which triples all the elements of a list of integers.

tripleAll : : $[$Integer$][$Integer$]$

Question Give a definition of a function which converts all small letters in a String into capitals, leaving the

other characters unchanged.

capitalize : : String $$ String

Question Define the function below, which returns the list of divisors of a positive integer $($and the empty

list for other inputs$).$ For instance, divisors $12=[1,2,3,4,6,12]$

divisor : : Integer $[$Integer$]$

Question A prime number $n$ is a number whose only divisors are $1$ and $n.$ Using divisors or otherwise define

a function which checks whether or not a positive integer is prime $($and returns False if its input is not a

positive integer$).$