How many of them? Or is it even possible?
1 2 0 4 1 < s r < 7 9 2 7
How many pairs of integers ( r , s ) with 0 < s < 4 0 0 satisfy the above inequalities?
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1 solution
How you generated first inequality?
@Monu Kumar Consider 1 2 0 4 1 − 1 2 0 x + 7 9 y 4 1 x + 2 7 y . It gives 1 2 0 ( 1 2 0 x + 7 9 y ) y ( 4 1 × 7 9 − 1 2 0 × 2 7 ) < 0 as 1 2 0 4 1 < 7 9 2 7 .
Thank you sir
[partial result]
It is not difficult to show that 1 2 0 4 1 < 1 2 0 x + 7 9 y 4 1 x + 2 7 y < 7 9 2 7 for all positive integers x , y .
As question requires 0 < s < 4 0 0 , we have 0 < 1 2 0 x + 7 9 y < 4 0 0 , which is true all positive integer pairs ( x , y ) = ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) ( 2 , 1 ) , ( 2 , 2 ) . Hence it is easy to check that 1 9 9 6 8 , 2 7 8 9 5 , 3 5 7 1 2 2 , 3 1 9 1 0 9 , 3 9 8 1 3 6 are the possible answers for s r .