To find the value of
n in the expression
1000!=3n×m where
m is an integer not divisible by 3 , we need to determine the largest power of 3 that divides 1000 !. This can be accomplished by using Legendre's formula, which is also known as de Polignac's formula. The formula is used to determine the exponent of a prime number
p in the factorization of
n !
Legendre's formula is given by:
where
⌊x⌋ represents the floor function, which gives the greatest integer less than or equal to
x, and
p is the prime number we are interested in (in this case, 3).
We need to find:
Let's calculate each term step by step:
‌⌊‌⌋=⌊333.3333⌋=333‌⌊‌⌋=⌊111.1111⌋=111‌⌊‌⌋=⌊37.037⌋=37‌⌊‌⌋=⌊12.3457⌋=12‌⌊‌⌋=⌊4.115⌋=4‌⌊‌⌋=⌊1.371⌋=1 For powers higher than
3∧6, the terms will be zero since 1000 divided by any higher powers of 3 will be less than 1 .
Summing these values, we get:
e3(1000!)=333+111+37+12+4+1=498Therefore, the value of
n is:
n=498The correct option is:
Option A