This one is a very very simple one for nay one in class xi-xii but ffor the sub xi guys..
Write 105 as teh sum of consecutive integers in 8 ways...
Dont just give the answer but also think why it works!
1:o
it turns out to be a hyperbola equation
and we have to find the lattice points on hyperbola [12]
62hmm.. that to me seems so non intuitive..
I am sure there is a better and much more simpler way to think if you could do something better..
Waise I am sorry if the impression i gave by the comment in the question was that it is intuitive to figure out why there are 8 solutions!
62dude.. you are far from it!
again sorry if i gave a wrong impression..
waise on the second thought, you have come very close to the explanation to why there are 8 solutions :)
62@rpf yup
@soumik... why have u ignored -ve factors,, and y have u taken 105 not 210?
1so the problem boils down ro two break 2*3*5*7 in two factors
by combinotrics its easy to see the numbers of ways is
p=2x3y5z7w
p'=2a3b5c7d
so it is simultenous solution of integral equations
so answer is 24=16
so i guess not 8 but 16 solutions exist for positive integres
so on total 32 solutions [11]
11Ultimate thing is to solve
2nx+n(n-1)x=210.. n is a natural, x is an integer.
62@rpf.. you are coming closer to the answer... first look yes 32 should be the answer.. but think again of what the two terms stand for? :)
1didnt get
sir , i think u have wrongly seen that they are consequitive
they are x-y and x+y-1 ! [1]
62no i havent wrongly seen that..
see what i am trying to say is that
what do x-y and x+y-1 represent?
1{105}
{52,53}
{34,35,36}
{19,20,21,22,23}
{12,13,14,15,16,17,18}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14}
{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14}
{-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
{-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
{-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
there are more than 8 ways[4]
62okie. harsh.. goof up agreed [3]
but now the question is to find the number of ways and justify that number [4]
23RPF , i got ur approach wen u explained it to me in the chatbox ,but
it shud be \frac{x(x+1)}{2}-\frac{y(y-1)}{2}=105
where x = last term and y = first term ????
and so (x+y)(x-y+1)=210
it shud be dis way na ??[7]