N times 21 is a four digit number such that the first and last digits are a 1. Looking only at the last digit, we see that the 3rd digit must also be a 1 else the four digit number would not end in 1.

So we need to find a two digit number, ending in 1, such that 21 times it has the required properties.

So we try 11, 21,... I get that 91 times 21 is 1911. As there are only nine two digit numbers ending in 1 we can easily work out which is the one.

-Dan