2621) x=63 y=19 ; x-3y =6
1)If x2-y2-84y=2012,x,y belonging to N,find x-3y?
2)Let S={20,21,22,23,.....210).Consider all positive differences of elements of S.If M is the sum of all these differences,find the sum of the digits of M?
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10 Answers
262well i am not posting here since i have no proof, but only logic .
1057watever it may be....it shud be sumthing gud....u cant put values like 63 and 19 to check for the equation???
1057in question no 2 i figured out M= \sum_{r=0}^{10}{(2r-10)2^{r}}
but how to proceed after dis....do i really have to find the value of M and then sum its digits or is there a easy alternative??
11Well, I also got the answer to the 2nd question as 27. I actually took out the value of M. Its easy that way, but a bit long. Does anyone have a shorter alternative?
1x^2 = y^2 + 84y + 1764+ 248
x^2 = (y +42)^2 + 248
x^2 - (y+42)^2 = 248
x-y-42 * x+y+42 = 248
since we are looking for integral solutions, factorise 248
and then we can obtain values of x and y
262after ameyaloya's 2nd step,
i used the fact that n2 + (2n+1) = (n+1)2
thus adding 2n+1 to a perfect square we get another perfect square.
thus the remaining task is to divide 248 into two or more such numbers .
it can easily be seen that as 248= 123 + 125 = {2(61)+1} + {2(62)+1}
hence y+42 = 61 ; y=19 and x=62+1 =63