Question

The number $21!$ $= 51{,}090{,}942{,}171{,}709{,}440{,}000$ has over $60{,}000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

$\dfrac{1}{21}$

$\dfrac{1}{19}$

$\dfrac{1}{18}$

$\dfrac{1}{2}$

$\dfrac{11}{21}$

Answer: B

Concepts:Legendre’s Formula factor counting prime factorization

Difficulty rating: 1730

Solution:

The exponent of $2$ in $21!$ is $\lfloor 21/2 \rfloor$ $+ \lfloor 21/4 \rfloor$ $+ \lfloor 21/8 \rfloor$ $+ \lfloor 21/16 \rfloor$ $= 10 + 5 + 2 + 1$ $= 18.$ Every divisor has the form $2^i b$ with $0 \le i \le 18$ and $b$ odd; it is odd exactly when $i = 0.$ So the fraction of odd divisors is $\dfrac{1}{18 + 1} = \dfrac{1}{19}.$

Thus, the correct answer is B.

2017 AMC 12B Problem 16

Answer: B

Difficulty rating: 1730

Solution:

Problem 16 in Other Years