## 27. Large Families

The Martens have more children than the Duponts. Suppose the difference of the squares of these two numbers is 24 and that both families have more than one child. How many children do the Martens have?

70
27. Let $m$ and $d$ be the number of children in the Marten and Dupont families, respectively. According to the problem,

$$
m^{2}-d^{2}=24 \text {, i.e., }(m+d)(m-d)=24\left(=2^{3} \cdot 3\right) \text {. }
$$

Therefore, the following values for $m+d, m-d$, and thus $2 m[=(m+d)+(m-d)]$, $m$, and $d$ are possible:

| $m+d$ | $m-d$ | $2 m$ | $m$ | $d$ |
| :---: | :---: | :---: | :---: | :---: |
| 24 | 1 | 25 | $12 \frac{1}{2} ?$ | $11 \frac{1}{2} ?$ |
| 12 | 2 | 14 | 7 | 5 |
| 8 | 3 | 11 | $5 \frac{1}{2} ?$ | $2 \frac{1}{2} ?$ |
| 6 | 4 | 10 | 5 | $1 ?$ |

The only valid value, therefore, is $m=7$. The Martens have 7 children.

Remark. The first and third possibilities are not valid because the number of children must be an integer, and the last one is not valid because, according to the problem, each family has more than one child.