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This problem trains for: SAT-I, AMC-8, GMAT.

What is the smallest integer that gives a remainder of 1 when divided by 3 and a remainder of 11 when divided by 12?

Denote the desired integer number with a. If a gives a remainder of 1 when divided by 3, then it has the general form:

where k is some integer.

Similarly, a must also fulfill:

Since a is an invariant for this problem we can set up an equation. Now we have to find the integers k and p that fulfill the equation:

so that a is smallest.

However, from:

Dividing each side by 3:

On the left hand side we have a multiple of 3 and on the right hand side we have the number 1. This equality cannot be satisfied by any integer values k and p.

Therefore, there is no solution to this problem.