$77$ is multiplied by another two-digit number with identical digits. The third
digit of the product, counting from left to right, is a $3$. What is the product?
I just guessed $55$ and it turned out to yield the right answer ($4235$), but I was wondering if there was a more mathematical way to this. I actually did put the two-digit number as $(10a+a) = 11a$, which gave me $847a$, but without knowing how many digits there are in total in the product, how can I carry on from there?
From $847*a$, we get that the one's digit will be $7*a \mod 10$ which is congruent to $3$ only if $a = 9$, which would lead to a $4$-digit number with $3$ as the last and fourth digit. Therefore, your number must be 4 digits. The third digit, then, is $\frac{7*a - (7*a \mod 10)}{10} + (4*a \mod 10)= 3$
$$\frac{7*a - (7*a \mod 10)}{10} + (4*a \mod 10)= 3$$ $${7*a - (7*a \mod 10)} + (40*a \mod 10)= 30$$ $$7*a - (7*a \mod 10) = 30$$
Clearly, $5$ is the only solution.