Common factor of (43^43+47^43) & (43^47+47^47) - stepwise please?

Question

Question - 1. Common factor of [(43 to the power of 43)+(47 to the power of 43)] & [(43 to the power of 47)+(47 to the power of 47)]
Stepwise answer please

Question 2. If 7 to the power of 126 is divided by 48, find the reminder - Stepwise answer please

Answer 1

For the first question, recall the formula for factoring sums of the form x^(2n + 1) + y^(2n + 1):

x^(2n + 1) + y^(2n + 1) = (x + y)(∑((-1)^k * x^k * y^(2n - k))), where the sum is taken from k=0 to 2n. (This is just like the formula for factoring the sum of two cubes, except the second term goes on longer.)

Thus:

(43^43 + 47^43) = (43 + 47)(43^42 - 43^41 * 47 + ... - 43 * 47^41 + 47^42)

(43^47 + 47^47) = (43 + 47)(43^46 - 43^45 * 47 + ... - 43 * 47^45 + 47^46).

So 90 is a common factor. It happens that it is the greatest common factor, but I am not sure how to prove this without using a computer.

For the second question:

We can write 7^126 = (7^2)^63 = (49)^63.

Notice that, for any n, 49n = 48n + n. This means that 49n differs from n by a multiple of 48. Thus, if we divide 49n by 48, we get the same remainder as if we were to divide n by 48.

Thus, if we divide 49^63 by 48, we get the same remainder as if we divided 49^62 by 48; which is the same remainder as if we divided 49^61 by 48; and so on down, until we get the same remainder as if we divided 49 by 48.

Therefore, the remainder when dividing 7^126 by 48 is 1.

(If you know group theory: Work out 7^126 within Z/48Z, the cyclic group of order 48. The argument is easier.)

Answer 2

Most of these cities and states you named have a very high concentration of blacks, poor blacks to be exact. I live in Chicago, is one of the worst cities not because of president Barack Obama but because of the people and their government. Most of these cities have a high crime rate because the schools and the community have failed the people. I have lived in Minneapolis, MN and they are a wonderful city and state since they are for the women and children. They voted for pres Barack Obama also and they are very well kept and clean. I like Minneapolis because of the education system it's a woman and kid state they got tons of government services and resources. I also like the cultural diversity and the landscapes. The Mall of America in Bloomington, MN is too cool and so is Nicollet Mall in downtown Minneapolis. The people are friendly and they got so many people there from all over the world like Whites, Blacks, Hmongs, Native Americans, Indians, people from the caribbean and a whole lot of Somalians. I like Chicago because I was born and raised there and we got a lot to do here and our city is big and a tourist attraction. plus we have one of the best downtown's ever. The crime is bad the education is ok depending on where you live the hospitals are great and the housing authority here sucks. The cultural diversity is ok we got people from all walks of life here also, but they all live on the Northside of the city especially in Rogers Park. Minneapolis help you with everything thing from getting into a shelter and getting housing to getting an education and a job. Most cities with less crime have good schools diversity and good resources like Minnesota.

Answer 3

You put this in the wrong section. It should be in Jokes.

Answer 4

go to hell