Euclid 2020/Problem 2

Problem

(a) The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of m equals the product of the tens digit and ones (units) digit of $m$ , what is $m$ ?

(b) Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1 : 4$ . How many gold marbles should she add to change this ratio to $1 : 6$ ?

(c) Suppose that n is a positive integer and that the value of $\frac{n^2 + n + 15}{n}$ is an integer. Determine all possible values of $n$ .

Solution

(a) Perform casework on the ones digit since it must be odd. If it is $1$ , then the tens and hundreds digit will be the same, which is not permitted. If it is $3$ , then $m=623$ is valid. Otherwise, the tens digit will be $1$ , so digits will not be distinct; thus $\boxed{m=623}$ .

(b) Since $1:4=20:80$ , she has $20$ and $80$ black and gold marbles, respectively. Since $1:6=20:120$ , she would need $120$ gold marbles to create the new ratio, so she must add $120-80=\boxed{40}$ gold marbles.

(c) $\frac{n^2 + n + 15}{n}=n+1+\frac{15}{n}$ , so all that is necessary is for $\frac{15}{n}$ to be an integer. This is achieved when $n$ is a positive factor of $15$ ; namely $\boxed{n=1,3,5,15}$ .

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Euclid 2020/Problem 2

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