N 311: Fill & Download for Free

We seek[math]S = \big\{n \in \mathbb [/math][math]N[/math][math]: [/math][math]311[/math][math] \equiv 23 \bmod{n}, 419 \equiv 23\bmod{n}\big\}[/math].Thus [math]n[/math][math][/math] divides both [math]311–23=288[/math] and [math]419–23=396[/math], and [math]n>23[/math]. Since [math]288=2^5 \cdot 3^2[/math] and [math]396=2^2 \cdot 3^2 \cdot 11[/math], [math]n[/math][math][/math] must be of the form [math]2^a \cdot 3^b[/math], with [math]a, b \in \{0,1,2\}[/math]. The only such [math]n>23[/math] is [math]36[/math].Hence [math]S=\{36\}[/math]. [math]\blacksquare[/math]

With single-payer healthcare (Medicare or Medicaid for all), the saving would be entirely selective. Why? Because single-payer would remove the single biggest barrier to healthcare cost that exists today - regulatory capture.By moving to single-payer, the Government effectively negotiates with providers, pharmaceutical companies, hospitals and medical device manufacturers using bulk delivery of 311 million patients. Just do the simple math on any procedure:$10,000 / N=1 = $10,000$10,000 / N=311 million = $0.00003215434No one is advocating N=1 - but what we've arrived at through the years is trending in that direction - and it's loosely called "selective health coverage."Single-payer by definition is the 2nd equation because its based on "universal health coverage."What we have in the U.S. is effectively the worst of all possible worlds when it comes to population health. We are often distracted by individual stories of heroic success - but that is NOT population health.The U.S. system is largely based on an artifact of WW II - employer sponsored insurance (about 150 million Americans). Through the years - we've tried to adjust the model for the elderly (over 65 = Medicare), the poor (Medicaid), and Vets (about 8M), but it has not worked. The effect on population health - as evidenced by this one chart - is a global embarrassment and a continuing National crisis:

Millions would die - unnecessarily.The assumption - which is popular - is that insurance is the root cause of what ails our healthcare system. It is. But not in the way that many people think it is.There are about 200 countries on the planet – of which only about 40 have a formal healthcare system. Of those 40 ‒ the U.S. is the only one without universal coverage.What we've had for the last 60+ years is a system of "selective" health coverage. At the risk of oversimplification, let's take a procedure that "costs" $10,000.$10,000 / n=1 = $10,000 (direct cost)$10,000 / n=311 million = $0.00003215434Health care insurance coverage has the most dramatic effect on cost - when everyone is covered.Unfortunately, for the last 60 years - what we've had in the U.S. is "selective" health coverage - largely based on either employment - or ability to pay. Last year the Commonwealth Fund released a report on findings from a 2012 study. That headline reflected the full toll of our unstated policy of "selective" health coverage):84 Million People Were Uninsured for a Time or Underinsured in in 2012- The Commonwealth FundThat's just a staggering number of people - basically 1/3 of the non-elderly (under 65) population of the country.Where it gets messy is the typical connection between "universal coverage" and "single payer." They should not be confused - but often are - and typically in pursuit of a political agenda (based on fear of "socialized" healthcare).The fact is - It’s entirely possible ‒ and often desirable ‒ to keep those two models separate. Universal coverage refers only to who gets access to healthcare services ‒ not how it is delivered ‒ or paid for. That is a seismic distinction.A single payer "system" - by default - is universal coverage (U.K., Canada)Universal coverage, however, is often "multi-payer" (Germany)The scientific basis for universal coverage is this. The demand for healthcare services will always outstrip supply (globally). The only economic mechanism we have to deal with that disparity ‒ is healthcare insurance. The whole point of healthcare insurance is to use actuarial science ‒ at scale ‒ to distribute total healthcare costs as fairly (and evenly) as possible ‒ across an entire population.If you continually erode the size of that population – the cost spiral can only go up. At a theoretical point, the cost of healthcare insurance would be equal to the cost of the service (slightly higher because administrative fees would be additive). The reason universal coverage works (and why all the other industrialized countries have adopted it) is because it maximizes the distribution of cost. In maximizes the denominator and you really do want that.Footnote: Zeroing out health insurance will have a very nominal effect. We know this because the net profits of healthcare insurance companies is relatively small:.. and ... relative to our National Healthcare Expenditure (NHE - currently running over $3 trillion per year) .. insurance profits don't even register: