Study guides

☆☆

Q: What is 20.47 as a mixed number in lowest terms?

Write your answer...

Submit

Still have questions?

Continue Learning about Math & Arithmetic

According to Mersenne 2n - 1 is a prime number(where n is also a prime number).As we all know the first prime number is 2 so putting the value of 2 in Mersenne's expression we get 22 - 1 = 3. So, 3 is the first Mersenne prime.Mersenne expression was considered as a method of finding primes. But it didn't always give prime number. Let us consider an example:Putting n = 11 in the expression we get 211 - 1 = 2047, but 2047 is not a prime number.Visit the below related link to know more.

The largest integer is 211 - 1 which is 2048 - 1 = 2047

On Converting between Arabic and Roman Numbers we get that : mmxlvii means 2047 where M=1000 , xl=40 and vii=7.

First let us define a perfect number. We call a number perfect if it is a positive integer and is the sum of all it proper divisors. The proper divisors part means we do not include the number itself even though any number divides itself. So for example 6 is perfect because it is a positive integer and 1+2+3=6. Not we excluded the number 6 even though 6 does divide 6.Now we could look at any number and find all the divisors of the number except the number itself. Next add them all and if the sum is equal to the number itself. If so then that number is a perfect number. While this method will always work, it is quite tedious.Euclid found another way thousands of years ago. He discovered that you could find the first four perfect numbers by using the formula 2p-1(2p − 1) where p is a prime such as 2,3,5 or 7. So let us consider the first prime 2 and plug it in Euclid's formula. 22-1 (22 -1)is equal to 21 (4-1)=2x3=6 which is the first perfect number. It is important to notice that the (2p − 1) portion is always giving us a prime number. The perfect number we would obtain with the primes 2,3,5, and 7 are also even. For example 23 -1=7 which is prime and we multiply that by 22 so 4x7=28 which is perfect and even.Everything was great until we tried the 5th prime which is the number 11. When we plug that into the (2p − 1) part just as we did 2 and 3, we get the number 211 − 1 = 2047 = 23 × 89 so it is not a prime number like the values we get with 2,3,5, and 7. Why do we care about the fact the this is not prime? We were looking for perfect numbers not primes?Because when we multiply 2047 by 211-1 we find out the the number we get is not perfect.So Euler was correct only for those first few values.When (2p − 1) is prime it has a special name. It is called a Mersenne prime named after Marin Mersenne who was a monk in the 17th ( a prime number by the way) century who studied math and primes and a field known as number theory. In order for (2p − 1) to be prime is has been found that it is necessary but not sufficient for p to be prime. In math a necessary condition is one that must hold true for the statement to be true. So p must be a prime. However a sufficient condition is one that says if this is satisfied the statement is always true. So the prime 11 showed us the (2p − 1) is not always prime and this means the condition is not sufficient. However, if p is not a prime (2p − 1) will not be prime either.A century later, Euler proved that all perfect numbers are generated by (2p-1 )(2p − 1).Euclid has tried but could not prove it. We can find the first 39 perfect numbers by using the primes: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917Note that all these number are even. We do not know if there are any odd perfect numbers. It has been shown that there are no odd perfect numbers in the interval from 1 to1050 but we can't say in general.We don't even know how many perfect numbers or Mersenne primes there are. While we suspect the number is infinite, that has not been proven yet.So here are the first few perfect numbers and you will see they become quite larger very fast!6,28,496,8128,33550336,8589869056,137438691328,2305843008139952128,2658455991569831744654692615953842176,191561942608236107294793378084303638130997321548169216,13164036458569648337239753460458722910223472318386943117783728128,14474011154664524427946373126085988481573677491474835889066354349131199152128,23562723457267347065789548996709904988477547858392600710143027597506337283178622239730365539602600561360255566462503270175052892578043215543382498428777152427010394496918664028644534128033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646976,141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128,54162526284365847412654465374391316140856490539031695784603920818387206994158534859198999921056719921919057390080263646159280013827605439746262788903057303445505827028395139475207769044924431494861729435113126280837904930462740681717960465867348720992572190569465545299629919823431031092624244463547789635441481391719816441605586788092147886677321398756661624714551726964302217554281784254817319611951659855553573937788923405146222324506715979193757372820860878214322052227584537552897476256179395176624426314480313446935085203657584798247536021172880403783048602873621259313789994900336673941503747224966984028240806042108690077670395259231894666273615212775603535764707952250173858305171028603021234896647851363949928904973292145107505979911456221519899345764984291328You can find every know Mersenne prime and hence every know perfect number on a site called GIMPS. I have included a link. But even more than seeing them, you can help discover the next one and become quite famous!Now this nice answer on perfect numbers will conclude with an important theorem.If 2k-1 is a prime number, then 2k-1(2k-1) is a perfect number and every even perfect number has this form. I will include a proof by Chris Caldwell. It uses the sigma function which is a function that finds the sum of the divisors. I will give link for the sigma function for those who are interested.Proof: Suppose first that p = 2k-1 is a prime number, and set n = 2k-1(2k -1). To show n is perfect we need only show sigma(n) = 2n. Since sigma is multiplicative and sigma(p) = p+1 = 2k, we knowsigma(n) = sigma(2k-1)(sigma(p)) = (2k-1)2k = 2n.This shows that n is a perfect number.On the other hand, suppose n is any even perfect number and write n as 2k-1m where m is an odd integer and k>2. Again sigma is multiplicative sosigma(2k-1m) = sigma(2k-1)(sigma(m)) = (2k-1)(sigma(m)).Since n is perfect we also know thatsigma(n) = 2n = 2km.Together these two criteria give2km = (2k-1)(sigma(m)),so 2k-1 divides 2km hence 2k-1 divides m, say m = (2k-1)M. Now substitute this back into the equation above and divide by 2k-1to get 2kM = sigma(m). Since m and M are both divisors of m we know that2kM = sigma(m) > m + M = 2kM,so sigma(m) = m + M. This means that m is prime and its only two divisors are itself (m) and one (M). Thus m = 2k-1 is a prime and we have prove that the number n has the prescribed form.It is even easier to prove that if for some positive integer n, 2n-1 is prime, then so is n.So this question about perfect numbers has generated some interesting discussion of Mersenne primes and included some exciting number theory. This is a beautiful topic in math that is often not covered.

For people paid every two weeks, there are 26 pay periods in a year. Two weeks makes 10 business days, for a total of 260 working days, not accounting for vacation time that you take. In the US people commonly say there are 260 weekdays/work days in a yearor 2080 work hours in a year based on 52 weeks per year.(260 = 5wkdys/wk*52wks)(2080=40hrs/wk*52wks/yr)The number can vary by year. For example, 2012 had 261 weekdays and 366 total days. Few people work every single weekday so the "work day" count is skewed when you take into account holidays that happen to fall on weekdays (up to 6 major US ones per year), vacation time, and sick time.Note: Federal and State employees typically have 10 holidays per year with an 11th one occurring every 4th year (for the Presidential Inauguration) regardless of whether they fall on weekends or not. If their holiday falls on a Saturday then it is recognized on Friday and if it falls on Sunday then it is recognized the next Monday.Weekdays per year: 260-262.Weekdays minus holidays falling on weekdays: 255-258.Years with 262 weekdays:1908 1912 1920 1924 1936 1940 1948 1952 1964 1968 1976 1980 1992 1996 2004 2008 2020 2024 2032 2036 2048 2052 2060 2064 2076 2080 2088 2092Years with only 260 weekdays:1905 1910 1911 1916 1921 1922 1927 1933 1938 1939 1944 1949 1950 1955 1961 1966 1967 1972 1977 1978 1983 1989 1994 1995 2000 2005 2006 2011 2017 2022 2023 2028 2033 2034 2039 2045 2050 2051 2056 2061 2062 2067 2073 2078 2079 2084 2089 2090 2095Years with only 255 weekdays minus holidays falling on weekdays (AWESOME!)1900 1901 1902 1905 1906 1907 1911 1913 1916 1917 1918 1919 1922 1923 1929 1930 1933 1934 1935 1939 1941 1944 1945 1946 1947 1950 1951 1957 1958 1961 1962 1963 1967 1969 1972 1973 1974 1975 1978 1979 1985 1986 1989 1990 1991 1995 1997 2000 2001 2002 2003 2006 2007 2013 2014 2017 2018 2019 2023 2025 2028 2029 2030 2031 2034 2035 2041 2042 2045 2046 2047 2051 2053 2056 2057 2058 2059 2062 2063 2069 2070 2073 2074 2075 2079 2081 2084 2085 2086 2087 2090 2091 2097 2098

Related questions

These numbers will go evenly into 2,047: 1, 23, 89, 2047.

2,394.99

odd

1 23 89 2047.Its factors are: 1, 23, 89 and 2047

If leading 0s are permitted then 0247 is smallest. If not, 2047

2047

The 1st of February 2047 will be on a Friday.

1/3 x 2047 = 682.3 recurring (that is, 682.3333...)

December 25 is on a Wednesday in 2047.

There are 5280 feet in one mile. Therefore, 2047 feet is equal to 2047/5280 = 0.3876893 recurring (that is, 0.38768939393...) miles.

The phone number of the Silent Wings Museum is: 806-775-2047.

Yes.

People also asked

Featured Questions