1986 AIME Problems/Problem 8
Problem
Let be the sum of the base logarithms of all the proper divisors (all divisors of a number excluding itself) of . What is the integer nearest to ?
Solution
Solution 1
The prime factorization of , so there are divisors, of which are proper. The sum of multiple logarithms of the same base is equal to the logarithm of the products of the numbers.
Writing out the first few terms, we see that the answer is equal to Each power of appears times; and the same goes for . So the overall power of and is . However, since the question asks for proper divisors, we exclude , so each power is actually times. The answer is thus .
Solution 2
Since the prime factorization of is , the number of factors in is . You can pair them up into groups of two so each group multiplies to . Note that . Thus, the sum of the logs of the divisors is half the number of divisors of (since they are asking only for proper divisors), and the answer is .
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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