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Q: A counterexample to "$k\mid n$ if and only if $k\mid n^2$" I am looking to prove the following statement false:
Let $k$ be a positive integer, then $k\mid n$ if and only if $k\mid n^2$.
So I am trying to find a $k$ where this does not hold but after aimlessly trying numbers I was curious if there is a methodical way to find $k$.

A: $12$ does not divide $6$, but it divides $6^2=36$.

A: $9\mid 12^2$ but $9\nmid 12$. ${{{{{{{{{{}}}}}}}}}}\qquad{{{{{{{{{{}}}}}}}}}}$

A: The methodical way would be to notice that this is in fact true, if $k$ is a prime number.
Then take the smallest non-prime (greater $1$), $k=4$. Then it is not far to see that $4$ does not divide $n=2$, but it does divide $n^2=4$.

A: One could even take $k = n^{2}$ for $n > 1$. Then we'd have that $k|n^{2}$ but $k$ does not divide $n$.

A: Hint $\,\ k\mid n^2\!\iff\! k\mid n\,\ $ is true $\iff k\,$ is squarefree (see there for many characterizations)
For if $\,k\,$ is not squarefree then $\, k = ab^2,\,\ b\nmid 1,\,$ hence $\,k\mid (ab)^2\,$ but $\,k\nmid ab\,$ (else $\,b\mid 1).$
Conversely if $\,k\,$ is squarefree then every prime $\,p_i\,$ in the unique prime factorization of $\,k\,$ occurs to the power $\,1.\,$ By Euclid's Lemma $\,p_i\mid k\mid  n^2\,\Rightarrow\,p_i\mid n,\,$ therefore $\,k = {\rm lcm}\{\,p_i\}\mid n,\,$ by existence & uniqueness of prime factorizations (or by lcm = product for coprimes)


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New York Plaza Hotel Takes a Turn On the Block
Written By Gretchen Chapman | March 25, 2016 | Published in Neighborhood News, Real Estate Market Trends
As of March 16th, New York's iconic luxury Plaza Hotel, will go on the auction block next month. According to Bloomberg Business, "Billionaire brothers, David and Simon Reuben hold the mortgage on the five-star hotel and have scheduled a foreclosure auction for April 26."
Next month's auction falls in the shadow of the scandal of previous owner, the Indian Financial Group, Sahara India Pariwar. Currently, Sahara India's founder and chairman, Subrata Roy, is serving time in a jail in Tahir, India. Subrata Roy faces charges as the result of 3.9 million dollar refund suit in India. Unable to make bail, Roy attempted to leverage the Plaza and the London's Grosvenor House, but still remains in custody.
New ownership is hardly a new concept for the Plaza Hotel, as it has changed hands several times over the course of its 110 year history. Most notably, presidential candidate, Donald Trump, who purchased the hotel in 1988 for 407 million dollars. However, with his ownership came a controversial remodel, where parts of the building came out "chintzy." Also came a swift change of economic tides in the early 90's, and as the real estate market went downhill, Trump sold the Plaza to avoid personal bankruptcy.
Despite its longstanding presence in New York, the Plaza is an expensive property, making it difficult for owners to keep long-term. That being said, Donald Trump's short lived ownership of the Plaza symbolized his place in old-money New York history.
The Plaza first opened its doors on October 1st, 1907, and was originally intended to be a residence hotel for wealthy New Yorkers. However, nightly visitors could rent rooms for as little as $2.50 a night, keeping in mind decades of monetary inflation, of course. The Plaza's conception came from the minds of financier, hotelier and business owner: Bernhard Beinecke, Fred Sterry, and Harry S. Black. They spent an "unprecedented" 12 million dollars on construction of the 19 story building. As of today, the Hotel's website includes a list of common amenities like "white glove butler service, 24 carat gold plated and white marble vanities." Not to mention, three in-house spas and uncommonly large rooms by New York Standards.
Perhaps you remember seeing the Plaza in Alfred Hitchcock's film suspense, crime thriller North by NorthWest? Or in the classic Eloise children's books, where the main character, a confident six-year-old named Eloise, lives at the "tippy-top floor" of the luxury residence. For decades, the Plaza, hung a portrait rendering of the fictional character on the walls of the lobby.
The Plaza's history and fantastical interiors represents a fantasy and aspirational New York lifestyle. However, its luxury facade does not reveal the behind the scenes jostling for its acquisition.
Manhattan Market Analysis
Midtown North

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MLS# 185020, 284 Barrataria Drive, St Augustine | Listings | Bosshardt Realty Services, LLC.
Waterfront gem in Barrataria Island! This 4 bedroom 2.5 bath home has everything to enjoy salt life living. Frontage on the Matanzas River and canal allows for ample watercraft or fishing opportunities. There is an existing boat house and 8500 lb. lift with floating docks and potential for more dockage. The home has been completely and tastefully remodeled in a coastal beach style you will love. The first level is open and spacious with beautiful distressed wood look, vinyl plank flooring.


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Video: Dark Skies Clear Way for Great Stargazing at Yosemite
A personal journey along one of the most spectacular coastlines in the world featuring the wildlife and wild places that make it so special.
Swirl into the world of maple syrup with Vivian Howard. Take a trip to Maine's Fryeburg Fair, discover the magic of caterpillars and learn more about the birds soaring across New England skies each fall.
Take a fresh look at our fragile planet and see just how much it's changing.
Tea, Tincture and Oil: The Making of Traditional Plant Medicine
After tending to and collecting clippings from medicinal plants in the wild, the Syuxtun Plant Mentorship Collective meets to make traditional medicines from purple sage, we'wey and olive oil.
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Paul R. Williams' Signature Architectural Style
Paul R. Williams not only worked on high-end design, he also worked on civic projects that carry his signature style.
The Future Fulfilled? Modernism's Effect on the California We Know Today
In the early post-1945 period, California served as the emblem of the American dream, the ideal of modernity and purveyor of modernism, which it broadcast to the nation and larger world. Did it manage to live up to its promise?
WPA Mural Studies
Are six large paintings part of a mural project commissioned by the WPA?
A Glimpse of Prohibition-Era L.A. at King Eddy Saloon
Explore the world beneath King Eddy and see how alcohol got into L.A. during Prohibition.
Video: Dark Skies Clear Way for Great Stargazing at Yosemite | KCET
Zach Behrens
Zach Behrens was formerly KCET's Director of News, Region and State, working on digital and on-air news products that relate to Southern California and beyond. He was also the Digital Executive Producer for "SoCal Connected."
Yosemite Nature Notes - 19 - Night Skies, by Yosemite National Park
For several years now, Steven Bumgardner has been producing "Yosemite Nature Notes," a video series that takes one aspect of Yosemite National Park and dives in for a short documentary. Late last month he launched his 19th episode, an awe-inspiring look at the night sky.
"It's been really satisfying," he said when reached by phone last week. And it should be: it took three patient summers of shooting, a total of 30 nights, including a half dozen star parties at Glacier Point atop Yosemite Valley. "Go an entire night; if you're lucky, you only get one or two shots."
All that work is paying off, too. Bumgardner, a former Sequoia National Park ranger and cave guide, says this has been the fastest growing "Nature Notes" in terms of views. And then there's something better: human reactions like the commenter who said, "This? actually made me cry a little bit."
And that may be part of the point: teach, inspire, engage, and ultimately create stewards out of the public so that Yosemite can continue to be protected for generations to come.
While the park boundary is probably safe from encroaching interests, the fate of seeing a clear dark sky is outside of its control. With inevitable long-term growth of California's fifth largest city, Fresno due south, comes more light. Bumgardner says Yosemite is doing pretty good -- unlike the challenges Death Valley National Park faces with neighboring Las Vegas -- but an issue not addressed in this documentary, light pollution's affects on plants and animals (and even air quality), is still important. "It's one of the environmental problems we could solve overnight."
If this video inspires you to check out Yosemite's dark skies, Bumgardner recommends Glacier Point and anywhere in Tuolumne Meadows since our eyes do not operate at their best past 8,000 feet (both are located at about 7,200 feet). Night sky viewing is optimal during and around a new moon on a clear night.
Related: The 7 Best SoCal Desert Locations for Night Sky Viewing
Say hello: "Like" SoCal Wanderer on Facebook and follow @SoCal_Wanderer on Twitter to talk about the latest in outdoors with other enthusiasts.
Anacapa to Ojai
Rosey visits Anacapa Island and the Ventura Harbor, rides her bike to Ojai, and checks out Thacher Observatory.
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Alana joined Endeavor in March 2015 and is the Director of Events & Partnerships. In this role, she supports Endeavor's corporate and foundation partnerships and manages Endeavor's global events including International Selection Panels and the signature Endeavor Gala. Prior to joining Endeavor, Alana was Assistant Director of an entrepreneurial non-profit working to leverage the power of the financial community to invest in the fight against cancer.
Alana received her BA from Elon University with a double major in International Studies and Political Science with a minor in Italian Studies. As a Massachusetts native, she remains a big Boston sports fan despite now living in NYC. She also enjoys hosting dinner parties, tackling home improvement projects, and serving as a mentor with Big Brothers Big Sisters of NYC.


Economics education: The development of activities and organizations to support effective teaching of economics at the high school and introductory level in higher eduction.
Cooperatives: The development of consumer cooperatives in the U.S. and other countries, Recently reviewed learning activities appropriate for high school educators.
Economic development: The focus is on South Africa in the context of university based outreach projects and education.


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2021 Pediatric Specialty Center Lectures
01/01/2021 - 8:30am EST to 12/31/2021 - 10:30am EST
Webinar Live
Care of the Pediatric Patient With Diabetes 01/05/2021 - 8:30am to 10:30am EST
POSTPONED to new Date: Seizures and Epilepsy; An Overview 02/02/2021 - 8:30am to 10:30am EST
Genetic Disorders 03/02/2021 - 8:45am to 10:45am EST
Infectious Disease 07/13/2021 - 8:30am to 10:30am EDT
Care of the Transgender and Gender Diverse Individual 08/24/2021 - 8:30am to 10:30am EDT
Pediatric Hematology/Oncology 09/28/2021 - 8:30am to 10:30am EDT
Neuromuscular Disorders 10/05/2021 - 8:30am to 10:30am EDT


Interstitial cystitis (IC) is a chronic health condition that causes pain and pressure in the bladder and pelvic area. Because of this, it is sometimes called painful bladder syndrome. IC can cause an urgent need to urinate, a frequent need to urinate, or both.
If you need help selecting bladder-friendly foods, there's an app for that. The ICN Food List, available for Android and iPhone, provides a reference guide you can use while shopping and eating out. The app contains a database with foods divided into three categories: bladder-friendly foods, foods worth trying cautiously and foods to avoid.


The meaning b ehind the land of misfit toys, and how events in my life have paired it with our current generation (millennials -present) goes back five years ago. Read as much, or as little as you'd like, the next few paragraphs simply give you a glimpse into my past, and the meaning behind The Land of Misfit Toys.
2012 was the first time my heart was ever shattered, not once, but consecutively over the span of four months. The first girl I ever loved left me, my Granny and Uncle both passed away and my wrist fractured into a million pieces. By telling you this I am not seeking pity, all I can hope for is to relate to other misfits of our generation. The memories above caused me to spiral into a deep depression resulting in me dropping out of school. Anyone who has ever been profoundly depressed understands how the whole world becomes heavy, as though no matter how hard you try, it seems impossible to get out of bed and to live a "normal" life.
For a long time there was a repetitive cycle where I barely knew the difference from one day, to the next. The best way I can describe it is through a song called Tomorrow Is Today by Billy Joel. In the song he says, "I don't care to know the hour 'cause its passing anyway. I don't care to see tomorrow because I saw it yesterday." It was as though the days and hours blurred into one, and no matter what you do, nothing seems to break the endless pattern, so, I stopped trying. One day, eight months later, my father intervened in a firm, but well needed talking to. As a result I got up out of bed and decided that it was time I went back to school.
In The Generation Of Misfit Toys, I will be blogging about other people's misfit stories, thoughts and opinions, as well as my own. If you would like to have your story shared, you can contact me at; thegenerationofmisfits@gmail.com.