The odd numbers 1, 3, 5, 7, …, 99 are all multiplied together. What is the highest power of 3 that divides the product?

Only the ODD multiples of 3 contribute. Odd multiples of 3 up to 99 are $3\times 1,3\times 3$, …, $3\times 33$ with the second factor odd — seventeen of them. Odd multiples of 9 are $9\times 1,9\times 3$, …, $9\times 11$ — six. Odd multiples of 27 are 27 and 81 — two. Odd multiples of 81: just 81 — one. Adding the layers gives 17 + 6 + 2 + 1 = 26. The trap 48 is the exponent of 3 inside the whole of 99!, which also counts the even multiples 6, 12, 18, … that are simply not in this product. Answering 33 uses ⌊99/3⌋ and counts each multiple only once, and 17 stops after the first layer.