Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2007 iTest Problems/Problem 30"

(Solution to Problem 30)
m
 
Line 3: Line 3:
  
 
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>.  
 
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers <math>a, b</math>, and <math>c</math>, and recalled that their product is <math>24</math>, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than <math>25</math> with fewer than <math>6</math> divisors. Help James by computing <math>a+b+c</math>.  
 +
 +
==Solution==
  
 
Since the [[prime factorization]] of 24 is <math>2^3 \cdot 3</math>, there are only a few options that need to be tested.  [[WLOG]], let <math>a \le b \le c</math>, so the possible options are <math>(1,1,24)</math>, <math>(1,2,12)</math>, <math>(1,3,8)</math>, <math>(1,4,6)</math>, <math>(2,2,6)</math>, and <math>(2,3,4)</math>.
 
Since the [[prime factorization]] of 24 is <math>2^3 \cdot 3</math>, there are only a few options that need to be tested.  [[WLOG]], let <math>a \le b \le c</math>, so the possible options are <math>(1,1,24)</math>, <math>(1,2,12)</math>, <math>(1,3,8)</math>, <math>(1,4,6)</math>, <math>(2,2,6)</math>, and <math>(2,3,4)</math>.
Line 8: Line 10:
 
Two of the conditions are that the sum is a two digit even number and the sum is less than <math>25</math>.  The only options left are <math>(1,3,8)</math> with sum as <math>12</math> and <math>(2,2,6)</math> with sum as <math>10</math>.
 
Two of the conditions are that the sum is a two digit even number and the sum is less than <math>25</math>.  The only options left are <math>(1,3,8)</math> with sum as <math>12</math> and <math>(2,2,6)</math> with sum as <math>10</math>.
  
−
The last condition is that the number has less than <math>6</math> divisors.  Since <math>12</math> has <math>6</math> divisors and <math>10</math> has <math>4</math> divisors, James’s desired sum is <math>\boxed{\textbf{10}}</math>.
+
The last condition is that the number has less than <math>6</math> divisors.  Since <math>12</math> has <math>6</math> divisors and <math>10</math> has <math>4</math> divisors, James’s desired sum is <math>\boxed{10}</math>.
  
 
==See Also==
 
==See Also==

Latest revision as of 18:24, 10 June 2018

Problem

While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers $a, b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.

Solution

Since the prime factorization of 24 is $2^3 \cdot 3$, there are only a few options that need to be tested. WLOG, let $a \le b \le c$, so the possible options are $(1,1,24)$, $(1,2,12)$, $(1,3,8)$, $(1,4,6)$, $(2,2,6)$, and $(2,3,4)$.

Two of the conditions are that the sum is a two digit even number and the sum is less than $25$. The only options left are $(1,3,8)$ with sum as $12$ and $(2,2,6)$ with sum as $10$.

The last condition is that the number has less than $6$ divisors. Since $12$ has $6$ divisors and $10$ has $4$ divisors, James’s desired sum is $\boxed{10}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 29 		Followed by:
Problem 31
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems/Problem_30&oldid=95064"