document.write( "Question 1210203: We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5? \n" ); document.write( "

Algebra.Com's Answer #851596 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "The prime factorization of 20 is $\"%282%5E2%29%285%5E1%29\"$ , so the prime factorization of 20^20 is $\"%282%5E40%29%285%5E20%29\"$ .

\n" ); document.write( "If a positive divisor is a multiple of 5, then it contains at least one factor of 5.

\n" ); document.write( "The divisors of 20^20 that are NOT multiples of 5 are those that contain only factors of 2.

\n" ); document.write( "To count the number of positive divisors of a number, add 1 to each exponent in the prime factorization of the number and multiply the resulting numbers.

\n" ); document.write( "The number of divisors of 20^20 is $\"%2840%2B1%29%2820%2B1%29=41%2A21=861\"$

\n" ); document.write( "The number of divisors of 20^20 that contain only factors of 2 is $\"40%2B1=41\"$

\n" ); document.write( "The number of divisors of 20^20 that DO contain at least one factor of 5 is 861-41 = 820.

\n" ); document.write( "The probability that a random divisor of 20^20 is a multiple of 5 is $\"820%2F861=%2820%2A41%29%2F%2821%2A41%29=20%2F21\"$

\n" ); document.write( "ANSWER: 20/21

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