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I have the following problem

A Three digit natural number's cross sum amounts to $14.$ When you read the number from right to left and subtract $22$, you obtain double the original number. Also the middle digit is equal to the sum of the outer two digits. Find the number

I am not sure what a cross sum is, and have been unable to find resources on it. We have been learning about matrices and know what a cross product is... but not a cross sum.

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    $\begingroup$ Sum of digits is the only plausible interpretation. $\endgroup$ – barak manos Feb 14 '15 at 11:18
  • $\begingroup$ am i right in thinking this should be solved with matrices? as per the second hint the ordering of the numbers is important. So that fits with matrices... $\endgroup$ – Arden Feb 14 '15 at 11:23
  • $\begingroup$ Well, you can solve it either with a $3\times3$ matrix, or with plain math over $3$ equations in $3$ variables... $\endgroup$ – barak manos Feb 14 '15 at 11:31
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You have the following equations (under the restriction of $0\leq{a,b,c}\leq9$):

  • $a+b+c=14$
  • $100c+10b+a-22=2(100a+10b+c)$
  • $b=a+c$

This is a system of $3$ equations in $3$ variables, which should be easy to solve:

  • $a=2$
  • $b=7$
  • $c=5$

Hence the number is $275$.

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  • $\begingroup$ @Arden: I moved the solution steps out of the answer, because inferring from your question, I'm pretty sure that you can handle the equation-system yourself. $\endgroup$ – barak manos Feb 14 '15 at 11:30
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If your question is just asking what a cross sum is, then a cross sum is just the sum of all digits of the given number.

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    $\begingroup$ Could you provide a reference, please? $\endgroup$ – Adrian Keister May 8 '18 at 17:14
  • $\begingroup$ What sort of reference would you like? Here's an example: The cross sum of 10 is 1+0 = 1 ; the cross sum of 275 = 2 + 7 + 5 = 14 (this happens to be the answer to the question above) . $\endgroup$ – ExtremeRaider Sep 21 '18 at 16:58
  • $\begingroup$ I'm talking about a reference to your definition. What you're referring to as the "cross sum" I'm more used to thinking of as the "digit sum". So if you could point to a mathematical dictionary or reference work of some kind defining "cross sum" the way you have it here, that'd be what I'm asking. $\endgroup$ – Adrian Keister Sep 21 '18 at 17:17
  • $\begingroup$ Well, I came across this word a few times in math class and also in a 'numberphile' video a long time ago, in case you're familiar with that. I haven't found any other reference. $\endgroup$ – ExtremeRaider Sep 21 '18 at 17:20

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