Write (on paper) the code for a RECURSIVE function called 'Icm' that: a. Takes two unsigned integers a and b as input, b. Returns an unsigned integer z as output which will be the least common multiple of a and b, c. Does any error checking or throws any exception as may be needed. A least common multiple z is the lowest number which can be fully divided by both inputs a and b without leaving a remainder. Then write the output of your function for the following two cases: (i) a = 97, b = 311 (ii) a = 77777, b = 67890
Write (on paper) the code for a RECURSIVE function called 'Icm' that: a. Takes two unsigned integers a and b as input, b. Returns an unsigned integer z as output which will be the least common multiple of a and b, c. Does any error checking or throws any exception as may be needed. A least common multiple z is the lowest number which can be fully divided by both inputs a and b without leaving a remainder. Then write the output of your function for the following two cases: (i) a = 97, b = 311 (ii) a = 77777, b = 67890
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![### Recursive Function for Calculating Least Common Multiple (LCM)
In this exercise, we aim to develop a recursive function `lcm` that:
1. **Takes Two Unsigned Integers as Input**
- The function will accept two unsigned integers, `a` and `b`, as its arguments.
2. **Returns an Unsigned Integer as Output**
- The function will return an unsigned integer `z`, which represents the least common multiple (LCM) of `a` and `b`.
3. **Error Handling**
- The function includes basic error checking to ensure it operates correctly or throws appropriate exceptions if necessary.
#### Definition of Least Common Multiple (LCM)
The least common multiple (LCM) of two integers `a` and `b` is the smallest positive integer `z` that is divisible by both `a` and `b` without leaving any remainder.
### Function Design
Here's the pseudocode for the recursive `lcm` function:
```python
def gcd(a, b):
if b == 0:
return a
else:
return gcd(b, a % b)
def lcm(a, b):
if a == 0 or b == 0:
raise ValueError("Inputs must be non-zero.")
return (a * b) // gcd(a, b)
```
### Explanation
1. **GCD Sub-function**:
- The `gcd` function calculates the greatest common divisor (GCD) using a recursive approach based on the Euclidean algorithm.
2. **LCM Calculation**:
- The `lcm` function computes the LCM using the relationship between GCD and LCM:
\[
\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}
\]
- It performs a check to ensure that neither `a` nor `b` are zero before proceeding.
### Test Cases
Utilize the function to compute the LCM for the following pairs:
1. **Case (i)**:
- Inputs: `a = 97`, `b = 311`
- Output:
\[
\text{LCM}(97, 311) = 30167
\]
2. **Case (ii)**:
- Inputs: `a = 77777`, `b = 678](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d2af6b8-35d3-4327-94c1-a05ff77beaf0%2F194dd31b-5a23-4ce8-8651-977673501deb%2F8m5hxu_processed.png&w=3840&q=75)
Transcribed Image Text:### Recursive Function for Calculating Least Common Multiple (LCM)
In this exercise, we aim to develop a recursive function `lcm` that:
1. **Takes Two Unsigned Integers as Input**
- The function will accept two unsigned integers, `a` and `b`, as its arguments.
2. **Returns an Unsigned Integer as Output**
- The function will return an unsigned integer `z`, which represents the least common multiple (LCM) of `a` and `b`.
3. **Error Handling**
- The function includes basic error checking to ensure it operates correctly or throws appropriate exceptions if necessary.
#### Definition of Least Common Multiple (LCM)
The least common multiple (LCM) of two integers `a` and `b` is the smallest positive integer `z` that is divisible by both `a` and `b` without leaving any remainder.
### Function Design
Here's the pseudocode for the recursive `lcm` function:
```python
def gcd(a, b):
if b == 0:
return a
else:
return gcd(b, a % b)
def lcm(a, b):
if a == 0 or b == 0:
raise ValueError("Inputs must be non-zero.")
return (a * b) // gcd(a, b)
```
### Explanation
1. **GCD Sub-function**:
- The `gcd` function calculates the greatest common divisor (GCD) using a recursive approach based on the Euclidean algorithm.
2. **LCM Calculation**:
- The `lcm` function computes the LCM using the relationship between GCD and LCM:
\[
\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}
\]
- It performs a check to ensure that neither `a` nor `b` are zero before proceeding.
### Test Cases
Utilize the function to compute the LCM for the following pairs:
1. **Case (i)**:
- Inputs: `a = 97`, `b = 311`
- Output:
\[
\text{LCM}(97, 311) = 30167
\]
2. **Case (ii)**:
- Inputs: `a = 77777`, `b = 678
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