Property 4
When determining the number of prime factors of an expression, the expression is divided by the prime number, and the result of the division is again divided by the prime number. This is continued until the division is smaller than the prime number. Finally all obtained quotients are added.
Example
\(20!=3^n\cdot a\)
\(a,n\in\mathbb{N}^{+}\)
For the maximum n value
| \(20\) | \(3\) | |
| \(-\;18\) | \(\enclose{circle}{6}\) | \(3\) |
| \(1\) | \(-\;6\) | \(\enclose{circle}{2}\) |
| \(0\) |
\(6+2=8\)
\(\max(n)=8\)
1.
\(a,n\in\mathbb{N}^{+}\)
\(20!=2^n\cdot a\)
\(\Rightarrow \max(n)=?\)
2.
\(a,n\in\mathbb{N}^{+}\)
\(15!=3^n\cdot a\)
\(\Rightarrow \max(n)=?\)
3.
\(a,n\in\mathbb{N}^{+}\)
\(\samefrac{18!}{5^n}=a\)
\(\Rightarrow \max(n)=?\)
4.
\(a,n\in\mathbb{N}^{+}\)
\(20\cdot21!=5^n\cdot a\)
\(\Rightarrow \max(n)=?\)
5.
\(a,n\in\mathbb{N}^{+}\)
\(30!=6^n\cdot a\)
\(\Rightarrow \max(n)=?\)
6.
\(a,n\in\mathbb{N}^{+}\)
\(\samefrac{28!}{15^n}=a\)
\(\Rightarrow \max(n)=?\)
7.
\(a,n\in\mathbb{N}^{+}\)
\(40!=9^n\cdot a\)
\(\Rightarrow \max(n)=?\)
8.
\(a,n\in\mathbb{N}^{+}\)
\(40!=8^n\cdot a\)
\(\Rightarrow \max(n)=?\)
9.
\(a,n\in\mathbb{N}^{+}\)
\(30!=12^n\cdot a\)
\(\Rightarrow \max(n)=?\)
10.
\(a,n\in\mathbb{N}^{+}\)
\(15!+16!=4^n\cdot a\)
\(\Rightarrow \max(n)=?\)
11.
\(a,n\in\mathbb{N}^{+}\)
\(20\cdot20!=2^n\cdot a\)
\(\Rightarrow \max(n)=?\)
12.
\(a,n,m\in\mathbb{N}^{+}\)
\(360\cdot30!=2^n\cdot3^m\cdot a\)
\(\Rightarrow \max(m+n)=?\)
13.
\(a,n\in\mathbb{N}^{+}\)
\(18!+19!=2^n\cdot a\)
\(\Rightarrow \max(n)=?\)