### into_the_world's blog

By into_the_world, history, 2 months ago,

In yesterday's AtCoder Beginner Contest 280 I didn't able to solve D. Then I was looking for solution and after seeing this submission I was shocked. Can anyone tell me how the logic work for this problem submission?

D — Factorial and Multiple

Submission

Thanks.

• 0

 » 2 months ago, # | ← Rev. 2 →   0 Suppose $k$ is a prime, then, the answer is always $k$. If $k$ is composite, remember that if a divisor an integer $n$ is less than or equal to $\sqrt{n}$, the other divisor is greater than or equal to $\sqrt{n}$. The solution uses that idea. The limit is $2 \cdot 10^{6}$, you can get a better limit by figuring out the last prime smaller than $10^{6}$(which is $999983$) and multiply by $2$. Why multiply by $2$? Because we can get a square of that prime as an input for $k$(which means that the answer would be $prime * 2$). Check the answer for $k = 1999966$ with $limit = 10^{6}$ to get what I mean and why the limit should be at least $1999966$.
•  » » 2 months ago, # ^ |   0 Thanks.
 » 2 months ago, # | ← Rev. 2 →   0 Here is an alternative approach. Suppose k is 1792. Convert 1792 into 2^8 * 7^1. It is clear that n needs to be at least 7 in order to cover the 7, but what is the n required to provide eight 2s? Let's count: 2 provides one 2. 4 provides two 2s. 6 provides one 2. 8 provides three 2s. Now we have (1+2+1+3 = 7) 2s. We still need one more. 10 provides one 2. Now we have eight 2s. This means n==10 is the max n needed. The other factor found earlier is n==7 but 7 is less than 10. So, the answer is 10. Edit: Just realized OP was referring to a particular submission.
•  » » 2 months ago, # ^ |   0 Thanks.