Largest Prime Factor

We can write $172!+171!$ as

$172\cdot(171!)+171!$

$\implies 173\cdot(171!)$

Since $173$ is prime and there are no greater prime factors than $173$ in $171!$ (by definition of a factorial), our desired answer is $\boxed{173}\ \blacksquare$

One should note that $n!=n\cdot(n-1)!$ So we then have

$(172!+171!)=(172\cdot 171!+171!)=171!\cdot(172+1)=171!\cdot173$

We have the largest prime factor as $\boxed{173}$ .

$(172!+171!)=171!(172+1)=171!\times 173$

Now, here we can see that 173 is clearly the largest prime factor and none of the factors of $171!$ can be as they are all less than 173. So, the answer is $\boxed{173}$

$172! + 171! = 171!(172 + 1) = 171!(173)$ . Anything on $171!$ is less than $173$ which is a prime since it's not divisible by any prime $\leq 13$ . Therefore, our answer is $\boxed {173}$ .

$172!+171!=171!(172+1)=171!\times173$ so largest prime factor is $173$

Note that $\begin{aligned} 172! + 171! &= 171! \cdot \left( 172 + 1 \right) \\&= 171! \cdot 173. \end{aligned}$ Note that $173$ is a prime number and the largest prime factor of $171!$ is less than $173$ .

Thus, the largest prime factor in question is $\boxed {173}$ .

$\blacksquare$

(172!+171!)

=171!(172+1)

=171!(173)

The above expression factors into $(172+1)(171!)$ , which is equivalent to $(173)(171!)$ . It is obvious that no single prime factor in $171 \times 170 \times 169 \times ...$ will be greater than $173$ (which is prime), so by default $\boxed{173}$ is the largest prime factor.

We're being asked to find something pertaining to the prime factors of a sum of factorials.. it's only natural to factor out the smaller one.

$(171! + 172!) = 171! \cdot (1 + 172) = 171! \cdot 173$

The largest prime factor of $171!$ is the largest prime $P \leq 171$ , which can be found to be $P = 167$ .

However, this is not the solution, as $173$ is in fact a prime number. Hence the answer is $\boxed{173}$

Notice that $172!+171! = (171!)(172+1) = 171! \cdot 173$ Since $173$ is prime, and the largest prime factor of $171!$ is less than or equal to $171$ , it is clear that the largest prime factor of $172!+171!$ will be $\fbox{173}$

172! + 171! = 172 * 171! + 171! = 171! * (172 + 1) = 171! * 173 = (1 * 2 * 3 * ... * 170 * 171) * 173 The largest prime factor, therefore, is 173.

[List of Primes] (http://en.wikipedia.org/wiki/List of prime numbers#The first 500 prime_numbers)