#2**+10 **

**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 \({\displaystyle n!} \)

see: https://en.wikipedia.org/wiki/Legendre%27s_formula

heureka
Mar 13, 2017

#1**+5 **

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

CPhill
Mar 13, 2017

#2**+10 **

Best Answer

**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 \({\displaystyle n!} \)

see: https://en.wikipedia.org/wiki/Legendre%27s_formula

heureka
Mar 13, 2017

#3**+10 **

**What is the largest integer $n$ such that $7^n$ divides $1000!$ ?**

**1.** We calculate 1000 in base 7:

\(\begin{array}{rcll} 1000_{10} &=& 2626_7 \\ \end{array} \)

**2**. The sum of the digits in base 7 is: \(\begin{array}{rcll} 2+6+2+6 = 16 \end{array} \)

**3.** \(n =\ ?\)

\(\begin{array}{|rcll|} \hline n &=&\frac{1000-(\text{sum of the standard base-p digits of 1000})}{7-1} \\ n &=&\frac{1000-16}{6} \\ n &=&164 \\ \hline \end{array} \)

\(7^{164} \) will be divide 1000! and \(7^{164} \) is one prime factor of 1000! for prime = 7.\(\)

heureka
Mar 13, 2017