1i have seen in MTG magzine many chinese ,candian, etc some quest r obj type
If 2n+1 and 3n+1 are both squares then prove that (n is a natural number):
1. 5n+3 is not a prime
2. n is divisible by 40
-
UP 0 DOWN 0 0 19
19 Answers
1
11Got me ?
There are so many others who are far far...(infinite times) better than me...
39hehe , good one soumik.
now that we have got u also here, we can start killing the maths ones
112) Since 2n+1 represents a square 2n+1=4s+1, meaning n is even, hence 3n+1=6s+1, thus x is odd. Using the fact that when a and b are odd, 8|a^2-b^2, we have 8|n… 2n+1 is actually congruent to 1,9,5 modulo 10, meaning n has to be congruent to 0,4,2,9,7 modulo 10 Checking for each, we have that n congruent to 0 (mod10) is the only possibility… Thu 5|n, meaning 40|n.
1links -
http://www.goiit.com/posts/list/algebra-prove-that-962062.htm
also see this if u like -
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=282100
9this had been dicussed in goiit already
i jus saw that thread there
9this has been finished in another forum
anyone here doing it ?
39but MTG is not all the source of all plympiads. real olympiads are those which need us to find the answer `NOT CHOOSE IT`
9oops ok
2nd part is not as easy as the 1st part
at the end of solution some manual calc needs to be done also for certain values
392nd part is quite good to think off.
so any juniors up to it?
(celestine i request u not to give any hint please)
39let 2n+1=k2 and 3n+1=m2
then 5n+3= (2k)2-m2 =(2k+m)(2k-m)
so obviously 5n+3 is not a prime
9hey manipal u need to prove n=40 X ()..
clue
5n+3 = 4(2n+1) - (3n+1)
