POTW - Week 4
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The Malone College Problem of the Week
The Malone College Mathematics Department proudly announces a weekly contest, where students such as you solve a mathematical puzzle and compete for fabulous merchandise! Each week, one person who correctly solves the problem will win a fabulous prize! This week’s prize…
a fin!
Yes, a fin, donated by the Department of Natural Sciences, proud sponsors of the Problem of the Week. If more than one person submits a solution, a random drawing will determine the winner. Authors of correct solutions will be posted the following Monday. Check out this space each Monday for a new puzzle! Actual odds of winning: 1 in n.
Problem of the Week #4:My Great-grandma (Gamma + 1) occasionally bakes some prime cookies!One way to prove that there are infinitely many primes is to go by way of contradiction, i.e. assume that there are finitely many primes, say p1, p2, ...,pn, and then prove that their product plus one is prime. What about any integer n? Is the product of the first n integers plus 1 a prime? The answer is no, as 4!+1=25 isn't prime. Find the first seven integers n for which n!+1 is prime. What does the title of this week's problem have to do with this computation? Solutions should be submitted to The Malone College Problem of the Week Headquarters in TS 243 by 3 p.m. Friday, October 27. The winner will be announced Monday, October 30.
The Problem of Last Week (#3)The sum of the first n integers is n(n+1)/2. If this is to divide n!, then n(n+1)/2 must divide n!. Clearly n will divide n!. But for n(n+1)/2 to divide n!, n+1 must be composite and greater than 2. Thus, if n+1 is an odd prime, n(n+1)/2 will not divide n!. There were five correct solutions submitted by James Glasgow, Lindsey Goldstein, Grant Johnson, Professor Rodd, and Dr. Williams. A three sided coin was tossed, and the winner is
James Glasgow. Congratulations James! You may pick up your miniature Lincoln portrait at The Problem of the Week Headquarters in TS 243. |