Question 29

Let $$P(n)$$ and $$S(n)$$ denote the product and the sum, respectively, of the digits of the integer $$n$$. For example, $$P(23) = 6$$ and $$S(23) = 5$$. Suppose $$N$$ is a two-digit number such that $$N = P(N) + 3S(N)$$. Then $$N = $$ ______.

Correct Answer: 55

Solution

Write $$N = 10a + b$$, so the condition becomes $$10a + b = ab + 3(a + b)$$, which simplifies to $$7a = b(a + 2)$$, that is $$b = \frac{7a}{a + 2}$$. Testing $$a = 1$$ to $$9$$, only $$a = 5$$ gives an integer, namely $$b = 5$$. So $$N = 55$$, and indeed $$25 + 3 \times 10 = 55$$.

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Let $$P(n)$$ and $$S(n)$$ denote the product and the sum, respectively, of the digits of the integer $$n$$. For example, $$P(23) = 6$$ and $$S(23) = 5$$. Suppose $$N$$ is a two-digit number such that $$N = P(N) + 3S(N)$$. Then $$N = $$ ______.

Solution

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