+26396 What is the largest integer $n$ such that $7^n$ divides $1000!$ ?
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!
see: https://en.wikipedia.org/wiki/Legendre%27s_formula
+130477 I'll take a stab at this one......
Notice that 1000! has 1000 / 7 = 142 numbers that are divisible by at least one 7
And 1000! has 1000/49 = 20 numbers that are divisible by 7^2 [ 20 additional 7s]
And it has 1000/343 = 2 numbers that are divisible by 7^3 [ 2 additional 7s ]
So.......the largest n such that 7^n will divide 1000! = (142 + 20 + 2) = 164
+26396 What is the largest integer $n$ such that $7^n$ divides $1000!$ ?
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!
see: https://en.wikipedia.org/wiki/Legendre%27s_formula
+26396 What is the largest integer $n$ such that $7^n$ divides $1000!$ ?
1. We calculate 1000 in base 7:
100010=26267
2. The sum of the digits in base 7 is: 2+6+2+6=16
3. n= ?
n=1000−(sum of the standard base-p digits of 1000)7−1n=1000−166n=164
7164 will be divide 1000! and 7164 is one prime factor of 1000! for prime = 7.
