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 count: add the number of multiples of 2, of 4, and of 8 up to 10. Multiples of 2: ⌊10/2⌋ = 5; of 4: ⌊10/4⌋ = 2; of 8: ⌊10/8⌋ = 1. The sum is 5 + 2 + 1 = 8, so x = 8.
As a cross-check, tally factors of 2 in each even term: 2 (one), 4 (two), 6 (one), 8 (three), 10 (one), totaling 1 + 2 + 1 + 3 + 1 = 8.
Stopping at multiples of 4 gives 7, the trap behind choice B; counting each even number once gives 5, the trap behind choice A.