self doubt

Question

the least no that must be subtracted from 1294 so that the remainder when divided by 9 ,11, 13 will leave in each case the same remainder

Answer 1

There is no explanation to this but the least number is 0

because 1294 divided 9,11,13 will have remainder 7

some interesting fact

1293 divided by 9,11,13 will have same remainder 6

1292 divided by 9,11,13 will have same remainder 5

similarly

1291 -> 4

1290->3

1289->2

1288->1

1287-> 0

reason

remainder will be either 0 or 1 or 2 .. 8 cannot exceed 8

now 9*11*13= 1287 so divide it by 9 or 11 or 13 remainder will be 0

1288 = 1287 +1 hence remainder will be 1

1289= 1287+ 2 hence remainder will be 2 and so on till 1294

another interesting thing is 1294 +1 =1295 will be last number ot this series to have common remainder which would be 8

Answer 2

LCM of 9 , 11 and 13 is 1287 so -
1287 % 9 == 0
also 1287 % 11 == 0
and 1287 % 13 == 0

so 1294 % 9  = 7
also 1294 % 11 = 7
and 1294 % 13 = 7

Since when 1294 id divided by 9 , 11 and 13 it gives same remainder as 7 .
Therefore 0 must be subtracted from 1294 so that the remainder must be same (7) when divided by 9 , 11 and 13 .

It is the standard method , although in this case the remainder is already same so we can directly go for the remainder by the given numbers 9 , 11 and 13 .