Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Integers and remainders.
The smallest integer that produces remainder of 2,4,6 and 1 when divided by 3,5,7,11 is
Integers
Remainders
Smallest integer
Answer: 419
B.Stat Objective Problem 85
Challenges and Thrills of Pre-College Mathematics by University Press
here 419=3(139)+2 and 419=7(59)+6
419=5(83)+4 and 419=11(38)+1
then required number=419.
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The smallest integer that produces remainder of 2,4,6 and 1 when divided by 3,5,7,11 is:
N=3*q1+2=3*(q1+1)-1=5*q2+4=5*(q2+1)-1=7*q3+6=7*(q3+1)-1=11*q4+1=
Considering only First Three N=3*q1+2=3*(q1+1)-1=5*q2+4=5*(q2+1)-1=7*q3+6=7*(q3+1)-1, we get N=105* Q-1
So ,N=105* Q-1=11*q4+1;
OR Q= (N+1)/105=4; q4= (N-1)/11=38;
so N=419