Difference between revisions of "1986 AIME Problems/Problem 8"

Revision as of 19:25, 9 April 2008

Problem

Let $S$ be the sum of the base $10$ logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000000$ . What is the integer nearest to $S$ ?

Solution

Solution 1

The prime factorization of $1000000 = 2^65^6$ , so there are $(6 + 1)(6 + 1) = 49$ divisors, of which $48$ 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 $\log 1 + \log 2 + \log 4 + \ldots + \log 1000000 = \log (2^05^0)(2^15^0)(2^05^1)\cdots (2^65^6).$ Each power of $2$ appears $7$ times; and the same goes for $5$ . So the overall power of $2$ and $5$ is $7(1+2+3+4+5+6) = 7 \cdot 21 = 147$ . However, since the question asks for proper divisors, we exclude $2^65^6$ , so each power is actually $141$ times. The answer is thus $S = \log 2^{141}5^{141} = \log 10^{141} = \boxed{141}$ .

Solution 2

Since the prime factorization of $10^6$ is $2^6 \cdot 5^6$ , the number of factors in $10^6$ is $7 \cdot 7=49$ . You can pair them up into groups of two so each group multiplies to $10^6$ . Note that $\log n+\log{(10^6/n)}=\log{n}+\log{10^6}-\log{n}=6$ . Thus, the sum of the logs of the divisors is half the number of divisors of $10^6 \cdot 6 -6$ (since they are asking only for proper divisors), and the answer is $(49/2)\cdot 6-6=141$ .

@@ Line 1: / Line 1: @@
 == Problem ==
-Let <math>\displaystyle S</math> be the sum of the base <math>\displaystyle 10</math> [[logarithm]]s of all the [[proper divisor]]s (all [[divisor]]s of a number excluding itself) of <math>\displaystyle 1000000</math>. What is the integer nearest to <math>\displaystyle S</math>?
+Let <math>S</math> be the sum of the base <math>10</math> [[logarithm]]s of all the [[proper divisor]]s (all [[divisor]]s of a number excluding itself) of <math>1000000</math>. What is the integer nearest to <math>S</math>?
+__TOC__
 == Solution ==
-The [[prime factorization]] of <math>1000000 = 2^65^6</math>, so there are <math>(6 + 1)(6 + 1) = 49</math> divisors (note the question asks for  proper divisors, so we ignore <math>1000000</math> and get <math>49-1=48</math>). The sum of multiple logarithms of the same base is equal to the logarithm of the products of the numbers.
+=== Solution 1 ===
+The [[prime factorization]] of <math>1000000 = 2^65^6</math>, so there are <math>(6 + 1)(6 + 1) = 49</math> divisors, of which <math>48</math> 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 <math>\log 1 + \log 2 + \log 4 + \log 5 + \ldots + \log 1000000 = \log 1 \cdot 2 \cdot 4 \cdot 5 \cdots = \log (2^05^0)(2^15^0)(2^05^1)(2^25^0) \ldots (2^65^6)</math>. Each power of <math>2</math> appears <math>7</math> times; and the same goes for <math>5</math>. So they appear <math>7(1+2+3+4+5+6) = 7 \cdot 21 = 147</math> times. However, since the question asks for proper divisors, we exclude <math>2^65^6</math>, so each number appears <math>141</math> times. The answer is thus <math>S = \log 2^{141}5^{141} = \log 10^{141} = 141</math>.
+Writing out the first few terms, we see that the answer is equal to <cmath>\log 1 + \log 2 + \log 4 + \ldots + \log 1000000 = \log (2^05^0)(2^15^0)(2^05^1)\cdots (2^65^6).</cmath> Each power of <math>2</math> appears <math>7</math> times; and the same goes for <math>5</math>. So the overall power of <math>2</math> and <math>5</math> is <math>7(1+2+3+4+5+6) = 7 \cdot 21 = 147</math>. However, since the question asks for proper divisors, we exclude <math>2^65^6</math>, so each power is actually <math>141</math> times. The answer is thus <math>S = \log 2^{141}5^{141} = \log 10^{141} = \boxed{141}</math>.
-== Solution 2 ==
+=== Solution 2 ===
 Since the prime factorization of <math>10^6</math> is <math>2^6 \cdot 5^6</math>, the number of factors in <math>10^6</math> is <math>7 \cdot 7=49</math>. You can pair them up into groups of two so each group multiplies to <math>10^6</math>. Note that <math>\log n+\log{(10^6/n)}=\log{n}+\log{10^6}-\log{n}=6</math>. Thus, the sum of the logs of the divisors is half the number of divisors of <math>10^6 \cdot 6 -6</math> (since they are asking only for proper divisors), and the answer is <math>(49/2)\cdot 6-6=141</math>.
@@ Line 13: / Line 15: @@
 == See also ==
 {{AIME box|year=1986|num-b=7|num-a=9}}
-* [[AIME Problems and Solutions]]
-* [[American Invitational Mathematics Examination]]
-* [[Mathematics competition resources]]
 [[Category:Intermediate Algebra Problems]]

1986 AIME (Problems • Answer Key • Resources)
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