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justification for using 0.75 as m/n ratio....
to produce min value of a+b......(153m-110n)/2 shud be minimum rite....nd 0.72<m/n<0.85......tell me the 2 evn integers such that they are in the ratio that differs frm 0.72 by the minimum and also the difference btween the two is minimum....=2 the minimum difference between 2 evn numbers(unequal)....u will say numbers like 6 and 8......(ratio 0.75) 10 and 8( ratio 0.8 but not closer to 0.72 than 0.75) u may say 10 and 12.....( ratio =0.833 but not as close to 0.72 as 0.75) in the same way u cud take a number of couples.....but u will find that the minimum difference btween 0.72 and the m/n ratio will only be achieved in the case of 6 and 8....if u go one step down taking 6 and 4...ull land up in truble as ratio is 0.66 which is out of the eqn...
so clearly the two integers m and n are 6 and 8 inserting which we get the min value of (a+b) as 28...
tell me whether its rite or wrong.....
Since a+11b is divisible by 13, so a-2b is also divisible by 13, hence 6a-12b is also divisible by 13
This implies that 6a+b is divisible by 13. ........(1)
Similarly, a+13b is div. by 11 implies that a+2b is div. by 11 which means that 6a+12b is div. by 11
Hence, 6a+b is divisible by 11. .............(2)
Now, (11,13) = 1
Thus, from (1) and (2), we have that 11*13 = 143 divides 6a+b.
So, 6a+b = 143k for some natural no. k.
Thus, 6a+6b = 143k+5b=144k+6b-(k+b)
This shows that 6 divides k+b and hence, k+b >=6 (both are nat. nos.)
So, 6a+6b = 143k + 5b = 138k+ 5(k+b) >=(138*1) + (5*6) = 168
thus, a+b>=168/6 =28
Hence, minimum value of (a+b) = 28
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my try.....
a + 11b=13m
a + 13b=11n
solving we have a= (169m-121n)/2
and b=(11n-13m)/2
and a+b =(156m-110n)/2...........1)
now as a>0 and b>0
we have..... 11/13> m/n>121/169
or 0.85 > m/n > 0.72
we take a favourable case of 0.75to get m and n as close to each other as possible so as to get the minimum value of the expression of 1)
due to which m/n=3/4.....m=3c and n=4c.......for integral values of a+b , c shud be even and 2 being the min evn number we have m=6 and n=8
using this we have min(a+b) as (936-880)/2 = 56/2 = 28....
so the minimum value of the expression (a+b) = 28.....