LCM or GCD（1）
组合数学（8）
容斥原理（2）
The Sports Association of Bangladesh is in great problem with their latest lottery 'Jodi laiga Jai'. There are so many participants this time that they cannot manage all the numbers. In an urgent meeting they have
decided that they will ignore some numbers. But how they will choose those unlucky numbers!! Mr. NondoDulal who is very interested about historic problems proposed a scheme to get free from this problem.
You may be interested to know how he has got this scheme. Recently he has read the Joseph's problem.
The Problem
There are N tickets which are numbered from 1 to N. Mr. Nondo will choose M random numbers and then he will select those numbers which is divisible by at least one of those M numbers. The numbers which are not
divisible by any of those M numbers will be considered for the lottery.
As you know each number is divisible by 1. So Mr. Nondo will never select 1 as one of those M numbers. Now given N,M and M random numbers, you have to find out the number of tickets which will be considered for
the lottery.
Each input set starts with two Integers N (10&=N&2^31) and M (1&=M&=15). The next line will contain M positive integers each of which is not greater than N. Input is terminated by EOF.
The Output
Just print in a line out of N tickets how many will be considered for the lottery.
Sample Input
Sample Output
题意：n个数中，有多少个数与m个数互质。
思路：只要先知道这m个数的倍数的个数，然后用总数减去就可以了。至于计算m个数的倍数的个数，用容斥原理解决。‘
#include &iostream&
#include &cstdio&
#include &cstring&
#include &vector&
ll num[20];
ll gcd(ll a,ll b){
else return gcd(b,a%b);
ll lcm(ll a,ll b){
return a/gcd(a,b)*b;
vector&ll&
int main(){
while(cin && n && m){
for(int i = 0; i & i++) cin && num[i];
for(int i = 1; i & (1&&m); i++){
vn.clear();
for(int j = 0; j & j++){
if(i&(1&&j)){
vn.push_back(num[j]);
if(vn.size()==1){
ans += n/vn[0];//cout&&ans&&
ll lc = vn[0];
for(int i = 1; i & vn.size(); i++){
lc = lcm(lc,vn[i]);
if(vn.size()&1){
cout&&n-ans&&
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(2)(6)(12)(35)(22)(22)(17)(13)(27)(6)(10)(1)(10)(9)(3)(24)The TSA Blog: More on the Liquid Rules:
Why We Do the Things We Do (Commenting Disabled)
Last week, there was a
blog by Jon Stokes, Senior Editor and Co-Founder, posing some questions on TSA’s liquids rules similar to other questions we’ve gotten on the blog so far. Kip Hawley wrote the following response, and we wanted to post it here for TSA blog readers to see as well.
Thanks for the question on liquids. We have lots of material on our site (tsa.gov) going into the liquids issue so that is available for background, including the video of it blowing up. I'll try here to break the question down into the sub-questions I hear most. I enjoy ars technica, especially that it is thoughtful and issue-oriented and I appreciate having the opportunity to address your question.
Was this a real threat? Yes, there was a very serious plot to blow up planes using liquid explosives in bombs that would have worked to bring down aircraft.
Why don't you just ban all liquids? Because our National Labs and international allies demonstrated to my satisfaction that there is, in fact, a scientific basis for allowing small amounts of liquids on as carry-on. We try to prohibit the minimum possible from a security standpoint. Also, the consequence of banning all liquids is a large increase in the number of checked bags, which creates its own issues.
Why can't multiple people bring on explosives in three-ounce containers and mix them post security? The tough one! Tough because there are parts of the reason that are truly classified but here goes... (read them all before throwing up your hands!)
We are involved in risk management. The question to me is: "What do you have to do to make a successful attack so complex that an intelligent enemy would recognize that the odds of success are too low?"
Because there are limits to our ability to detect every thing every time at the checkpoint, we use layers of security. For example, I and senior leaders at TSA work every day with the intelligence and law enforcement communities world-wide to get insights in how to make our security better -- frequently adding specific training and sometimes, respecting our obligations to the intell and law enforcement communities (like our remote control toys advisory), communicating directly to the public. Also, we reduce risk by a) adding behavior detection capability, K-9 teams, surge teams and documen and b) by undercover presence throughout the area behind the checkpoint, as well as better screening of the supply chain of items in the sterile area after the checkpoint.
We reduce risk by deciding what we believe is necessary for a completed bomb -- the core of the 100ml (3.4 ounce) limit. Extensive testing began the morning of August 10, 2006 -- the day the liquids plot was made public -- to determine if there is a level at which any liquid brought onboard a plane represents little risk. These were tests by multiple government agencies, National Laboratories and other nations and they assisted in the 3-1-1 formulation. We announced 3-1-1 on September 26, 2006 and that allowed travelers to go on overnight trips without having to check a bag. That is the trade-off: if 3-1-1 is too complicated, you can always just check your bag.
The preparation of these bombs is very much more complex than tossing together several bottles-worth of formula and lighting it up. In fact, in recent tests, a National Lab was asked to formulate a test mixture and it took several tries using the best equipment and best scientists for it to even ignite. That was with a bomb prepared in advance in a lab setting. A less skilled person attempting to put it together inside a secure area or a plane is not a good bet. You have to have significant uninterrupted time with space and other requirements that are not easily available in a secured area of an airport. It adds complexity to their preferred model and reduces our risk, having the expert make the bomb and give it to someone else to carry aboard. They are well aware of the Richard Reid factor where he could not even ignite a completed bomb. Simple is truly better for them. Also, bomb-makers are easier for us to identify than so-called clean 'mules.'
The container itself adds complexity. A 100ml container limits the effect of, and even the ability of, a detonation. It also forces a more precise mix, and a lot more boost -- which makes it easier to detect from that side. Even creative ways to smuggle liquids in are less effective because, eventually, they still have to mix it right and get it into the right container, etc. There are also issues with what kind of container you use, but let's leave them to puzzle that out further...
The baggie gives us two benefits: A) It serves as a visually identifiable, easy way to limit quantity. Even if they wanted to bring multiple bottles to mix, we limit the quantity of their total liquids as well (bottles "hidden" in the carry-on bag stick out). B) The baggie serves to concentrate the vapor - substances used to create liquid explosives are very volatile and emit fumes even through sealed bottles. (We have tested.) We have liquid explosives detectors that take advantage of the vapor concentration factor in the baggie. This way, we do not have to examine what's inside every bottle, regardless of what the label says.
The effect of pulling out liquids and aggregating them separately allows our security officers to have a clear look at the liquids -- and, perhaps just as important, it de-clutters the carry-on bag so that we have a clearer view of that as well.
With our medical exceptions, they have to talk to one of our Security Officers who can use a variety of methods to tell whether it presents a problem including test strips, and hand-held detectors that are highly effective, even with closed and sealed bottles. With the larger bottles, the other features needed to make it viable would be very apparent.
A few other points, this policy has been adopted in more than 80 countries worldwide and means that there are common rules almost everywhere you fly. The choice is a total ban or this, and we are working very hard at a technology solution that should make this better all around. Think early 2009 for that.
The challenge is to reduce risk on the things we know about (shoe bombs, liquids) while having enough other measures in place to disrupt what we don't know is coming. Any time we fixate on one thing, you have to be concerned about opening up something elsewhere. Balance, flexibility, and unpredictability are key. So is going on offense by being connected to intelligence / law enforcement and being proactive with our surge patrols, undercover activities, etc. AND getting TSA and passengers back on the same side! That last one is what we're trying to do at our checkpoint with our TSOs and online with our blog.
Whatever you think about our policies -- please recognize our Security Officers who train and test every day and will do whatever it takes to make you and your families safe when you fly. They are the best in the world
please give them a little recognition when you see them. Thanks for the opportunity to comment,
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Subscribe ToMisunderstanting the Million Dollar Curve - 推酷
Misunderstanting the Million Dollar Curve
Do you think that it's possible to use maths to &crack& the lottery code, to instantly win lottery games such as theUS Mega Millions and US Powerball?
Now, cryptographers from the Laboratoire de Math&matiques de Versailles in the University of Paris-Saclay have discovered something that may change the game of lottery altogether, and they're calling it the Million Dollar Curve.
Yet perhaps they should rename it to the Billion Dollar Curve as lottery jackpots are constantly hitting the billion-dollar point these days (thanks to rollovers).
Their aim is to find a convincing solution to the extreme &problem& of randomness. Here's what they have to say (and have fun trying to figure it out):
&Our method allows to build what we call a Publicly verifiable RNG [random number generator], from which we extract a seed that is used to instantiate and initialise a BlumBlum-Shub random generator. We then use the binary stream produced by this generator as an input to a filtering function which deterministically outputs secure and uniformly distributed parameters from uniform bitstreams,&
Essentially, to determine what the numbers are, the lottery number generator requires &seeds& (data) and while that data is easy to obtain, it's difficult to work with.
The numbers of the Million Dollar Curve are generated by a novel process and after enough seed information, it may one day produce actual lottery ticket predictions. In other words, this system is years, even decades away from being usable. The Million Dollar Curve is so powerful it is even able to (eventually) predict the outcome of the lottery even if there is random tampering from outside sources. For tampering to even be an issue, every single lottery draw across the world from the last draw must have been tampered with. You can read
, but it requires a degree in mathematics to even begin deciphering. Good luck!
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