A person takes at least one aspirin a day for 30 days. Suppose he takes 45 aspirin altogether. Is it possible that in some sequence of consecutive days, he takes exactly 14 aspirin?
The names of those who submit correct solutions will be posted on the Math Club bulletin board outside of the Math Department office and on the web page of the NIU Math Club. Small prizes may be awarded for correct solutions!
We were asked to show that if p > 5 is a prime number, then there exist positive integers a, b such that p-a2 is a proper divisor of p-b2.
(A proper divisor of an integer n is a divisor which is greater than 1 and less than n.)
Solution Interestingly, p doesn't have to be prime, exactly, but not any old integer will work. Here's one way to see what will work.
Write p as a2 + k with integers a and k, where k > 0 is chosen to be as small as possible. Then let b = |a - k|.
Then p - b2 = p - (a - k)2 = k + 2ak - k2, which is a multiple of k = p - a2. This shows that p-a2 is a divisor of p-b2; we need only check that it is a proper divisor. But the two can't be equal because b = |a - k| is clearly strictly smaller than a (Exercise: what did we assume that makes k nonzero? Also, why can't it be true that p-a2=1? Try replacing b with composite numbers, up to 100 say, and see what happens...)
We were asked to show that for a, b, c > 0, we have
Well for positive numbers a and b, we know a+b is positive and thus (a-b)2(a+b) is non-negative, or, stated a little differently,
Note that equality appears iff there are equalities in (1), that is, iff a = b = c.