How would you go about finding the lowest common multiple of the following numbers:

3, 7, 24, 86, 125, 214?

Go to this website for finding least common denominator, or all sorts of other things. CLICK HERE

269,640

but it seems awfully high. Is that right?

Well, AlisonWonder, I am afraid the correct answer is even higher than 269640 (which is not divisible by 86 or 125)...

The lowest common multiple of 3, 7, 24, 86, 125 and 214 is96621000   =   2³ ´ 3 ´ 5³ ´ 7 ´ 43 ´ 107.

I just realized that the link provided by Frankvan provides only the method which is applicable to small numbers like the ones in your question (which are not difficult to factor into primes).  Factoring larger numbers into primes is often very difficult, so it's not a realistic option (in fact some modern schemes in "public key" cryptography do rely on the fact that it's difficult to retrieve two large prime numbers from their product).

To find the least common multiple (LCM) of two large numbers, you're much better off computing first their greatest common divisor (GCD) with Euclid's algorithm (or related algorithms that are similarly efficient). Then use the relation:

LCM(a,b)   =   ( a ´ b ) / GCD(a,b)

Given huge numbers like:
a = 2562047788015215500854906332309589561
b = 6795454494268282920431565661684282819
This is how you would "easily" discover that:
LCM(a,b) = 15669251240038298262232125175172002594731206081193527869

Note: The above numbers (a and b) are not random ones.  They are both products of two very special 19-digit prime numbers.  (HINT: Their GCD is 1111111111111111111 and the other factors are of a similar nature...)

3, 7, 24, 86, 125, 214

Well, you could do it one by one.

3, 7 >>> 3*7 >>> 21
21, 24 >>> 3*7, 2*2*2*3 >>> 2*2*2*3*7 >>> 168
168, 86 >>> 2*2*2*3*7, 2*43 >>> 2*2*2*3*7*43 >>> 7224
7224, 125 >>> 2*2*2*3*7*43, 5*5*5 >>> 2*2*2*3*5*5*5*7*43 >>> 903000
903000, 214 >>> 2*2*2*3*5*5*5*7*43, 2*107 >>> 2*2*2*3*5*5*5*7*43*107 >>> 96,621,000

Darn you, DrGerard, you've beaten me again...