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Let x be the larger number and y be the smallest one. it says, x-y=1365---------(1) and also says x/y=6 leaving remainder 15.
x / y= 6  =>x= 6 * y--------(2)    
substitute equation 2 in equation 1, it'll become 
[6*y] -y=1365  {cauz value of x=6*y}   now, 6y-y=5y right?? so we'll say 
5y=1365------(3)  => y=1365/5 which is 273  therefore y= 273. Now the rest is easy!
substitute y in equation 2, that is x=6*y => x=6*273 and that is x=1638
now to confirm, 1638-273=1365 and divide x/y, you will get 6!!! hurray u got the answer. y is the smaller number=273