If 2ˣ is the greatest power of 2 that divides 10!, what is the value of x?
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Answer & Explanation
Correct answer
C
The exponent of 2 in 10! is found by Legendre's formula: add up the number of multiples of 2, of 4, and of 8 that are at most 10, since each such multiple contributes at least one extra factor of 2 beyond what the smaller power already counted. Multiples of 2 up to 10: ⌊10/2⌋ = 5. Multiples of 4 up to 10: ⌊10/4⌋ = 2. Multiples of 8 up to 10: ⌊10/8⌋ = 1. The sum is 5 + 2 + 1 = 8, so x = 8.
As a cross-check, tally the factors of 2 contributed by each even term in 10! = 1 × 2 × 3 × ⋯ × 10 directly: 2 contributes one factor of 2, 4 = 2² contributes two, 6 = 2 × 3 contributes one, 8 = 2³ contributes three, and 10 = 2 × 5 contributes one. The total is 1 + 2 + 1 + 3 + 1 = 8, matching the Legendre count.
Stopping the sum after multiples of 2 and 4 only, and missing the extra factor of 2 that 8 = 2³ contributes beyond the first two, gives 5 + 2 = 7. Counting each even number as contributing exactly one factor of 2, without noticing that 4 and 8 contribute more than one each, gives 5.