For any natural number n, F(n) is the least common multiple of all natural numbers upto. For how many values of n less than 100, F(n+1) - F(n) = 0?
Choose the correct choice from the following answers: A) 33 B) 64 C) 29 D) 27 E) 01
F(n) is the LCM of al natural numbers upto n.   For example,   F(3) = 6   F(5) = 60   F(6) = 60 etc.   F(n + 1) - F(n) = 0   ⇒⇒ F(n + 1) = F(n)   It would hapen only when factors of (n + 1) are already present in the LCM of first n natural numbers. In other words, Whenever (n + 1) is Prime number or is any Integral power of any prime number, F(n + 1) ≠≠ F(n)   for example, F(31) ≠≠ F(32) as 32 = 2525 which is, 2 raised to power 5. or F(4) ≠≠ F(5) as 5 is prime so the power of 5 does not contain in F(4).   Between 1 and 100, there are 25 prime numbers. So if (n+1) is one of these number then F(n + 1) ≠≠ F(n).   Also {4, 8, 16, 32, 64, 9, 27, 81, 25, 49} are prime numbers raised to some exponent.   Hence, none of these 35 numbers from 1 to 99 cannot be the possible values of n. So for   99 - 35= 
<gadget id="calculator">  99 - 35</gadget>
<output>64</output>
64 numbers F(n)= F(n+1)   Answer:64
<result>B</result>
For any natural number n, F(n) is the least common multiple of all natural numbers upto. For how many values of n less than 100, F(n+1) - F(n) = 0?