$A$ is counting the numbers from $1$ to $31$ and $B$ from $31$ to $1$. $A$ is counting the odd numbers only. The speed of both is the same. What will be the number which will be pronounced by $A$ and $B$ together?

### Concept & Logic This problem can be solved by modeling the spoken numbers as two simultaneous Arithmetic Progressions (AP). Since their counting speed is the same, we need to find the $n$-th term where both sequences produce the exact same value. ### Step-by-Step Solution * $A$ counts odd numbers starting from 1: $1, 3, 5, 7, \dots$ This is an AP with first term $a = 1$ and common difference $d = 2$. The $n$-th term for $A$ is $1 + (n - 1)2 = 2n - 1$. * $B$ counts backwards from 31 normally: $31, 30, 29, 28, \dots$ This is an AP with first term $a = 31$ and common difference $d = -1$. The $n$-th term for $B$ is $31 + (n - 1)(-1) = 32 - n$. * Since they pronounce the same number at the same time, we equate the two $n$-th terms: $$ 2n - 1 = 32 - n $$ * Solve for $n$: $$ 3n = 33 $$ $$ n = 11 $$ * Substitute $n = 11$ back into either formula to find the pronounced number. Using $B$'s formula: $$ 32 - 11 = 21 $$ ### Exam Strategy & Shortcut Use relative speed logic. $A$ moves $+2$ per step, and $B$ moves $-1$ per step. Their relative gap closes by $3$ units per step. The initial gap between their starting numbers ($31$ and $1$) is $30$. It takes $30 / 3 = 10$ steps *after* the first pronunciation to meet. So they meet on the $11$-th pronunciation. For $B$, stepping down 10 times from 31 lands directly on 21. ### Common Pitfall A common mistake is drawing out the entire table manually, which consumes excessive time and invites simple counting errors under exam pressure. Rely on the mathematical progression to guarantee accuracy. ### Final Answer Therefore, the correct answer is **21**.