Opposite-direction counting — synchronized calls Nitin counts down from 32 (32, 31, 30, …) while Sumit counts up only odd numbers starting at 1 (1, 3, 5, …), calling out numbers at the same speed. Will they ever call the same number at the same time?

Introduction / Context:We compare two synchronized sequences to test if a common term occurs at the same index.

Concept / Approach:At step t (starting t = 0 at the first call), Nitin says 32 − t. Sumit says 2t + 1. Set them equal and check for integer t ≥ 0.

Step-by-Step Solution:Equation: 32 − t = 2t + 1 ⇒ 31 = 3t ⇒ t = 31/3.t is not an integer ⇒ there is no synchronized step where both announce the same number.

Verification / Alternative check:Listing partial terms shows near misses but no exact match at the same step.

Final Answer:They will not call out the same number