Recall that the primes fall into three categories: Let Pi be the set of
primes congruent to 1 (mod 4) and P3 be the set of primes congruent to
3 (mod 4). We know that
{primes} = {2} UP, UP3.
We have previously proved that P3 is infinite. This problem completes
the story and proves that P1 is infinite. You can do this by following these
steps:
A) Fix n > 1 and define N = (n!)2 + 1. Let p be the smallest prime divisor
of N. Show p>n.
B) If p is as in part (a), show that p ⌘ 1 (mod 4). (To get started, note
that (n!)2 ⌘ 1(mod p), raise both sides to the power p1 2 and go from
there. You will need Fermat’s Theorem)
C) Produce an infinite increasing sequence of primes in P1, showing P1
is infinite.

(c) compare the results of parts (a) and (b). in general, how do you think the mode, median, and mean are affected when each data value in a set is multiplied by the same constant? multiplying each data value by the same constant c results in the mode, median, and mean increasing by a factor of c. multiplying each data value by the same constant c results in the mode, median, and mean remaining the same. multiplying each data value by the same constant c results in the mode, median, and mean decreasing by a factor of c. there is no distinct pattern when each data value is multiplied by the same constant. (d) suppose you have information about average heights of a random sample of airline passengers. the mode is 65 inches, the median is 72 inches, and the mean is 65 inches. to convert the data into centimeters, multiply each data value by 2.54. what are the values of the mode, median, and mean in centimeters? (enter your answers to two decimal places.) mode cm median cm mean cm in this problem, we explore the effect on the mean, median, and mode of multiplying each data value by the same number. consider the following data set 7, 7, 8, 11, 15. (a) compute the mode, median, and mean. (enter your answers to one (1) decimal places.) mean value = median = mode = (b) multiply 3 to each of the data values. compute the mode, median, and mean. (enter your answers to one (1) decimal places.) mean value = median = mode = --