Discussion
A set of S consists of,
i). All odd numbers from 1 to 55.
ii). All even numbers from 56 to 150.
What is the index of the highest power of 3 in the product of all the elements of the set SS?
Answer: A
Only the multiples of 3 in the set contribute, and each contributes as many threes as it contains.
Odd numbers from 1 to 55. The odd multiples of 3 are 3, 9, 15, 21, 27, 33, 39, 45, 51, carrying exponents 1, 2, 1, 1, 3, 1, 1, 2, 1:
\[ 1+2+1+1+3+1+1+2+1 = 13 \]
Even numbers from 56 to 150. These are the multiples of 6 in the range, 60 through 150, sixteen of them, with exponents 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1:
\[ 1+1+2+1+1+2+1+1+3+1+1+2+1+1+2+1 = 22 \]
\[ \therefore\; \text{index} = 13 + 22 = 35 \]
Note that 27 and 108 each supply three, and 9, 45, 72, 90, 126 and 144 each supply two; counting one per multiple would give 25 and miss the higher powers.
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