5. Each letter in the word MASHTOVIT represents a digit from the set $\{1,2,3,4,5,6,7\}$. Different letters represent different digits. The number $M A Ш TOBИT$ is odd and divisible by 3, and all consonants represent digits of the same parity. How many such numbers are there?

## Second Grade - B Category
5. In the number MASHTOVIT, each letter except $T$ appears exactly once ($T$ appears 2 times) and represents one of the digits from the set $\{1,2,3,4,5,6,7\}$. Therefore, the sum of the digits of the number MASHTOVIT is $1+2+3+4+5+6+7+T=28+T$. Since the number MASHTOVIT is odd, $T$ must be an odd digit. Furthermore, since the number MASHTOVIT is divisible by 3, the sum of its digits $28+T$ must also be divisible by 3, so $T=5$.

All consonants represent digits of the same parity, so we get that $M, Ш$, and $B$ are odd digits (since $T=5$ is an odd digit), i.e., $M, Ш, B \in \{1,3,7\}$. Since we have used all the odd digits, the vowels represent even digits, i.e., $A, O, И \in \{2,4,6\}$. Since different letters represent different digits, $M, Ш, B$ can be determined from the set $\{1,3,7\}$ in $3!=6$ ways, and $A, O, И$ from the set $\{2,4,6\}$ also in $3!=6$ ways, so the total number of such numbers is $6 \cdot 6=36$.

## Second Grade - B Category