Question 10

$$m$$, $$n$$, $$p$$ are all different natural numbers each between 2 and 9. If $$\frac{m + n + p}{m + n}$$ is an integer, then the number of all possible values of $$\frac{m + n + p}{m + n}$$ is

Solution

Write $$\frac{m + n + p}{m + n} = 1 + \frac{p}{m + n}$$, so $$m + n$$ must divide $$p$$. As $$m$$ and $$n$$ are different numbers greater than 2, $$m + n \ge 7$$, while $$p \le 8$$, so the only possibility is $$p = m + n$$, for instance $$(m, n, p) = (3, 4, 7)$$ or $$(3, 5, 8)$$. In every such case the value of the expression is $$1 + 1 = 2$$, so only 1 value is possible.

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$$m$$, $$n$$, $$p$$ are all different natural numbers each between 2 and 9. If $$\frac{m + n + p}{m + n}$$ is an integer, then the number of all possible values of $$\frac{m + n + p}{m + n}$$ is

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