Question 695366
<font face="Times New Roman" size="+2">


Let the hundreds digit be represented by *[tex \LARGE x], the tens digit be represented by *[tex \LARGE y], and the ones digit be represented by *[tex \LARGE z].


What we know to start:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 0\ <\ x\ \leq\ 4]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 0\ \leq\ y\ <\ 10]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 0\ \leq\ z\ <\ 10]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ z\ =\ 2x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ y\ +\ z\ =\ 14]


But since *[tex \LARGE z\ =\ 2x], we can write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ +\ y\ =\ 14]


The largest *[tex \LARGE x] can be is 4 so let's try that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3(4)\ +\ y\ =\ 14]


So if *[tex \LARGE x\ =\ 4], then *[tex \LARGE y\ =\ 2] and *[tex \LARGE z\ =\ 8] so the number is 428.


But *[tex \LARGE x] could equal 3.


So if *[tex \LARGE x\ =\ 3], then *[tex \LARGE y\ =\ 5] and *[tex \LARGE z\ =\ 6] so the number is 356.


But *[tex \LARGE x] could equal 2.


So if *[tex \LARGE x\ =\ 2], then *[tex \LARGE y\ =\ 8] and *[tex \LARGE z\ =\ 4] so the number is 284.


But *[tex \LARGE x] could equal 1.


So if *[tex \LARGE x\ =\ 1], then *[tex \LARGE y\ =\ 11] -- Oops!  That doesn't work because *[tex \LARGE y] must be less than 10.  *[tex \LARGE x\ =\ 0], doesn't work either because then you would only have a two digit number.


So the three possible answers are 428, 356, and 284.  All are less than 500, all have a units digit twice the hundreds digit, and all have 14 as a sum of digits.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>