为什么即使在较大范围内，此Python代码也不能为我提供解决方案？

Question

Below is the problem I'm trying to solve:

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?

It is the 5th problem from Project Euler ( http://projecteuler.net/problem=5 ). I wrote a code in Python to find out the number, but I'm unable to obtain a solution.

My code is:

def smallestNumber():
 smallest=0 ## Initializing the smallest number
 end=False ## Initializing the break condition for lower level For loop
 found=False ## Initializing the break condition for upper level For loop
 for i in range(21,10000000): ## Upper level for loop
    if i==10000000 and found==False: break ## This will break upper loop when range is satisfied
    for k in range(1,20): ## Lower level for loop
        if end==True: break ## This will break lower loop when range is satisfied
        if i%k==0: ## If condition to check whether current i value is divisible evenly by current k value
            if k==20: ## If k==20, this will make smallest equal to current i value and make both break conditions True
                smallest=i
                end=True
                found=True
            k=k+1
        else: ## if not divisible, this will increment upper level loop
            break
            i=i+1

 if found==False: print 'No value exists in this range'
 else: return smallest

(I'm new to stackoverflow and was unable to paste the actual code without messing up the formatting. I apologize for any inconvenience due to that).

I keep getting the output 'No value exists in this range' regardless of how big I make my range. I'm guessing that although my logic is alright, I have messed up the code somewhere since I'm a Python beginner.

It'd be great if someone can help me.

Thanks

Answer 1

Some things that are wrong:

1. The answer is greater than your upper limit of 10000000
2. Your should use xrange in Python 2, otherwise you'll have memory errors if you increase the upper limit
3. If you want all of the numbers from 1 to 20, you should use range(1, 21)
4. You should not manually increase loop counters, range or xrange do it for you

Answer 2

First of all, what you need is LCM of the numbers from 1 to 20. You can find better ways to find LCM than the one you've implemented in this code.

Next, there are a few mistakes here. I've edited your code to illustrate. Please see my code bellow, this shows the changes I had to make in order to make your code produce the correct result for finding the smallest number divisible by all numbers from 1 to 10:

def smallestNumber():
smallest=0 
end=False 
found=False
for i in range(21,1000000):
    if found==True: break # This will break upper loop when number is found
    for k in range(1,11): # the range is up to 11 here because we need to check numbers 1-10
        if end==True: break 
        if i%k==0:
            if k==10: ## 
                smallest=i
                end=True
                found=True
            #k=k+1 # not necessary
        else:
            break
            #i=i+1 # not necessary

if found==False: print 'No value exists in this range'
else: return smallest

As you can see, I had to comment out lines like i=i+1 because that is already being taken care of by using for i in range(21,10000000) . You would also notice that I had to change if i==10000000 and found==False: break to if found==True: break because checking for i=10000000 is not necessary and you should actually stop searching when your found is true. And that was the real bug in your code. Hope this helps.