Problem
"2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?"
Solution
To solve this problem, we need to understand the concept of the smallest multiple. A multiple of a number is a result of multiplying that number by any integer. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, and so on.
The smallest multiple that is divisible by a set of numbers is called the least common multiple (LCM) of those numbers. To find the LCM of two numbers, we can use the greatest common divisor (GCD) of those numbers. The relationship between LCM and GCD is given by the formula:
LCM(a, b) = (a * b) / GCD(a, b)
This formula can be extended to find the LCM of more than two numbers by iteratively calculating the LCM of pairs of numbers.
1. We
define a helper function, gcd,
which calculates the greatest common divisor (GCD) of two numbers using the Euclidean
algorithm.
2. Another
helper function, lcm,
calculates the least common multiple (LCM) of two numbers using the LCM formula
we discussed earlier.
3. The
smallest_multiple
function takes one argument, n,
which represents the upper limit (in this case, 20) for the range of numbers
for which we want to find the smallest multiple.
4. We
initialize result
to 1 as the starting point.
5. Using
a loop that iterates through all numbers from 1 to n, we update result by finding the LCM
of result
and the current number.
6. After
the loop finishes, result
will contain the smallest positive number that is evenly divisible by all
numbers from 1 to n.
Solving Project Euler Problems provides a great opportunity to practice coding and algorithmic thinking.
No comments:
Post a Comment