# what is a numbers cross sum?

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.

• Sum of digits is the only plausible interpretation. – barak manos Feb 14 '15 at 11:18
• 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... – Arden Feb 14 '15 at 11:23
• Well, you can solve it either with a $3\times3$ matrix, or with plain math over $3$ equations in $3$ variables... – barak manos Feb 14 '15 at 11:31

## 2 Answers

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 $275$.

• @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. – barak manos Feb 14 '15 at 11:30

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.

• Could you provide a reference, please? – Adrian Keister May 8 '18 at 17:14
• 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) . – ExtremeRaider Sep 21 '18 at 16:58
• 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. – Adrian Keister Sep 21 '18 at 17:17
• 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. – ExtremeRaider Sep 21 '18 at 17:20