Discussion
What is the highest power of 5 that divides the product \(1 \times 3 \times 5 \times ... \times 99\)?
Answer: B
We need to count the number of factors of 5 in the product of all odd numbers from 1 to 99. The multiples of 5 in this product are 5, 15, 25, 35, 45, 55, 65, 75, 85, 95.
Let's count the factors of 5 from each term:
5 = \(1 \times 5\) (one 5)
15 = \(3 \times 5\) (one 5)
25 = \(5 \times 5\) (two 5s)
35 = \(7 \times 5\) (one 5)
45 = \(9 \times 5\) (one 5)
55 = \(11 \times 5\) (one 5)
65 = \(13 \times 5\) (one 5)
75 = \(3 \times 25 = 3 \times 5 \times 5\) (two 5s)
85 = \(17 \times 5\) (one 5)
95 = \(19 \times 5\) (one 5)
Total number of factors of 5 is \(1+1+2+1+1+1+1+2+1+1 = 12\).
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