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Two algebra questions

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How many single-digit even natural number solutions are there for the equation $a+b+c+d = 24$ such that $a+b > c+d$?


marked as duplicate by Henning Makholm, Jonas Teuwen, anon, Zhen Lin, Martin Sleziak Jun 26 '12 at 15:59

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The single-digit even natural numbers are $2$, $4$, $6$, $8$. The sum is $24$, quite big. So the number of possibilities is not large. Almost any careful listing will do the job. But here is a possible systematic approach.

The average of our numbers must be $6$. If they are all $6$, we violate $a+b\gt c+d$.

So there is at least one $8$. There can't be three $8$'s, that makes the sum too big. So the number of $8$'s is $1$ or $2$.

Now listing should be straightforward. Do (i) one $8$ and (ii) two $8$'s.

(i) There is only one $8$, and the average is $6$, so we must have one $8$, two $6$'s, and one $4$. Where can they be? Because $a+b\gt c+d$, one of $a$ or $b$ must be $8$, and the other $6$. And therefore one of $c$ or $d$ is $6$ and the other $4$. List all cases. You should get $4$.

(ii) All yours!

  • $\begingroup$ I've forgotten: is $0$ widely considered a natural number or not? That is, I suspect this is a GMAT question - do they think that $0$ is a natural number? $\endgroup$ – davidlowryduda Jun 26 '12 at 15:42
  • 1
    $\begingroup$ @mixedmath: For someone in Logic, the answer is a clear yes. Some day the rest of the world may catch up. $\endgroup$ – André Nicolas Jun 26 '12 at 15:52

Here is one approach you can use:

The even single-digit (presumably in base 10) natural numbers are precisely 2, 4, 6, 8.

Thus, if $y$ and $z$ are such numbers, then $4\leq y+z\leq 16$, and of course $y+z$ is even.

Break the problem up as follows: find the number of solutions $w$, $x$ to $w+x=24$ such that $w$ and $x$ are even numbers such that $4\leq x<w\leq 16$, and then for each choice of $w$ and $x$, find the number of solutions to $a+b=w$ and the number of solutions to $c+d=x$ such that $a,b,c,d\in\{2,4,6,8\}$.


Here's a brute force approach:

L = (2,4,6,8)

count = 0
for a in L:
    for b in L:
        for c in L:
            for d in L:
                if (a+b+c+d)==24 and (a+b)>(c+d):
                    count = count + 1

print count

Which prints 11.


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