Difference between revisions of "1987 AIME Problems/Problem 3"
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By a proper [[divisor]] of a [[natural number]] we mean a [[positive]] [[integer|integral]] divisor other than 1 and the number itself. A natural number greater than 1 will be called ''nice'' if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | By a proper [[divisor]] of a [[natural number]] we mean a [[positive]] [[integer|integral]] divisor other than 1 and the number itself. A natural number greater than 1 will be called ''nice'' if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | ||
− | == Solution == | + | == Solution 1== |
Let <math>p(n)</math> denote the product of the distinct proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances: | Let <math>p(n)</math> denote the product of the distinct proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances: | ||
#It has exactly two distinct [[prime]] divisors. | #It has exactly two distinct [[prime]] divisors. | ||
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[[Sum]]ming these yields <math>\boxed{182}</math>. | [[Sum]]ming these yields <math>\boxed{182}</math>. | ||
+ | |||
+ | == Solution 2== | ||
Alternatively, we could note that <math>n</math> is only nice when it only has two divisors, which, when multiplied, clearly yield <math>n</math>. We know that when the [[prime factorization]] of <math>n = a_1^{b_1} \cdot a_2^{b_2} \cdot a_3^{b_3} . . . \cdot a_m^{b_m}</math>, the number of factors <math>f(n)</math> of <math>n</math> is <cmath>f(n) = (b_1 + 1)(b_2 +1)(b_3 +1) . . . (b_m +1).</cmath> | Alternatively, we could note that <math>n</math> is only nice when it only has two divisors, which, when multiplied, clearly yield <math>n</math>. We know that when the [[prime factorization]] of <math>n = a_1^{b_1} \cdot a_2^{b_2} \cdot a_3^{b_3} . . . \cdot a_m^{b_m}</math>, the number of factors <math>f(n)</math> of <math>n</math> is <cmath>f(n) = (b_1 + 1)(b_2 +1)(b_3 +1) . . . (b_m +1).</cmath> | ||
Since <math>n</math> is nice, it may only have <math>4</math> factors (<math>1</math>, <math>n</math>, <math>p</math>, and <math>q</math>). This means that <math>f(n) = 4</math>. The number <math>4</math> can only be factored into <math>(2)(2)</math> or <math>(4)(1)</math>, which means that either <math>b_1 = 1</math> and <math>b_2 = 1</math>, or <math>b_1 = 3</math>. Therefore the only two cases are <math>n = pq</math>, or <math>n = p^3</math>. | Since <math>n</math> is nice, it may only have <math>4</math> factors (<math>1</math>, <math>n</math>, <math>p</math>, and <math>q</math>). This means that <math>f(n) = 4</math>. The number <math>4</math> can only be factored into <math>(2)(2)</math> or <math>(4)(1)</math>, which means that either <math>b_1 = 1</math> and <math>b_2 = 1</math>, or <math>b_1 = 3</math>. Therefore the only two cases are <math>n = pq</math>, or <math>n = p^3</math>. | ||
+ | And then continue. | ||
== See also == | == See also == |
Latest revision as of 08:39, 29 July 2020
Contents
Problem
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
Solution 1
Let denote the product of the distinct proper divisors of . A number is nice in one of two instances:
- It has exactly two distinct prime divisors.
- If we let , where are the prime factors, then its proper divisors are and , and .
- It is the cube of a prime number.
- If we let with prime, then its proper divisors are and , and .
We now show that the above are the only two cases. Suppose that another nice number existed that does not fall into one of these two categories. Then we can either express it in the form (with prime and ) or (with ). In the former case, it suffices to note that .
In the latter case, then .
For , we need .
Since , in the case does not work.
Thus, listing out the first ten numbers to fit this form, . Summing these yields .
Solution 2
Alternatively, we could note that is only nice when it only has two divisors, which, when multiplied, clearly yield . We know that when the prime factorization of , the number of factors of is
Since is nice, it may only have factors (, , , and ). This means that . The number can only be factored into or , which means that either and , or . Therefore the only two cases are , or . And then continue.
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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