underworld-blood-war (2016)

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Published: June 22, 2022 (1 week ago)
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underworld-blood-war (2016)


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Movies title: Underworld: Endless War. Genre: Action. Year: 2011. Director: Oscar Wigmore.
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Ari Melber

06/20/12 01:00 PM EDT

Today is National Girls & Women in Sports Day.

Bloomberg:

On Monday, World Cup soccer is back, and the tournament has an unprecedented number of female players. More than half of the 32 teams entered this year’s competition are led by women. Last year, the competition was held in the U.S. for the first time; next year, the World Cup will be hosted in South Africa for the first time. Between now and the 2013 Women’s World Cup, America will host Olympic games and host the Women’s World Cup as well. There will be multiple women as captains at each of these events.

The gender of the players on the field and the coaches in the coaching box may seem like a cultural change, but it’s more than that. Inequality is an enduring fact, but women’s participation in sports in the U.S. has steadily increased over the past 40 years. Women’s hockey has grown from 2,000 people playing in the U.S. in 1985, to 15,000 in 2010. Female players in the U.S. Basketball Association shot nearly 60 percent over the same period. According to a 2010 study by Harvard University, women now comprise 47 percent of high school boys’ sports players; 20 percent of the girls’ and women’s high school basketball teams is female. Studies that track the growth of both girls and women in sports are consistent in finding substantial increases. For example, one study concluded that 80 percent of American high school female athletes would not even think of participating in track and field in 1920.

Perhaps the biggest factor driving women’s participation in sports is a wider cultural

Underworld: Endless War.Q:

Proving the absence of a prime $p$ with $p$ not dividing the squarefree part of $n$

I’m currently doing the following problem:

If $n$ is not divisible by any prime $p$ with $pmid 2$, prove that there is no prime $p$ with $p$ not dividing the squarefree part of $n$.

Here is my strategy so far:

Let $n=2^{r_1} cdot 3^{r_2} cdot ldots cdot p^{r_p}$ be the prime factorization of $n$. If $r_1>0$ prove that $pmid2^2$ (i.e. $pmid1$). This is obviously wrong as we can take $p=5$.
If $r_1=0$ then $p=2$ and it is ok as we can take $p=2$.
If $r_1=0, r_2>0, ldots, r_p>0$ or $r_2=0, r_3>0,ldots,r_{p-1}>0$, then we can firstly find a prime $p$ with $pmid2$ and secondly $p$ must divide $2^{r_2} cdot 3^{r_3} cdot ldots cdot p^{r_p}$.
If $r_1=r_2=ldots=r_{p-1}=0$ then $p=3$ is a prime with $pmid2$ and so we can also prove the statement.

However, I am stuck on the case $r_1=0, r_2=0, ldots, r_{p-1}>0$ or $r_2=0, r_3=0, ldots,r_{p-2}>0$. It seems like this should be easy but I’m not having any success with the proof.
How can I prove this statement? Is my proof valid?
Edit: If $n=2^r cdot p^s$ with $pmid2$ then $pmid2^r cdot p^{s-r}$ and
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