if your friend has sufficiently mathematical knowledge there are ways to convince your friend by using a computer program :-)
Some time ago i saw a similiar problem where the numbers were percentages https://groups.google.com/group/de.rec.denksport/browse_thread/thread/aa7dde6a6043d7c4.
So the solution does not have to be an integer. Then one can formulate the problem as linear programming problem. The meaning of the variable $x1110$ for example is: it is the percentage of soldiers that have the first property "lost a left leg" (first index of variable $x$ is
$1$), the second property "lost a right leg" (second index of variable $x$ is $1$), the third property "lost a left arm"
(third index of variable $x$ is $1$) but not the fourth property "lost a right arm" (fourth index of variable $x$ is $0$). The four statements paritions the set of all soldiers in $2^4$ subsets and the following holds:
the percentages are between 0 and 100 (c1 ... c32) , the percentages of soldiers with the first,...,fourth property is 70,75,80,85 (c33,...,c34) and the sum of all percentages is 100. we are interested in the case where the number of soldiers that have lost all four limbs is minimal (min). We can use this input directly in the linear equation solver applet I found at http://vinci.inesc.pt/lp/ and get the expected solution. If all calculation are done in exact arithmetic (I think the applet does not) then 10 is the minimal integer solution.
c17: x0000 <= 100;
c33: x1000 + x1001 + x1010 + x1011 + x1100 +x1101 + x1110 +x1111 = 70;
c34: x0100 + x0101 + x0110 + x0111 + x1100 +x1101 + x1110 +x1111 = 75;
c35: x0010 + x0011 + x0110 + x0111 + x1010 +x1011 + x1110 +x1111 = 80;
c36: x0001 + x0011 + x0101 + x0111 + x1001 +x1011 + x1101 +x1111 = 85;
c37: x0000 + x0001 + x0010 +x0011 + x0100 + x0101 + x0110 +x0111 + x1000 + x1001 + x1010 +x1011 + x1100 + x1101 + x1110 + x1111 = 100;
x1111 = 10.0
x0111 = 30.0
x1011 = 25.0
x1101 = 20.0
x1110 = 15.0
all other variables (percentages) are 0