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Difference between revisions of "Euclid 2020/Problem 2"

(Created page with "2. (a) The three-digit positive integer <math>m</math> is odd and has three distinct digits. If the hundreds digit of m equals the product of the tens digit and ones (units)...")
 
 
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==Problem==
 
(a) The three-digit positive integer <math>m</math> is odd and has three distinct digits. If the
 
(a) The three-digit positive integer <math>m</math> 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
 
hundreds digit of m equals the product of the tens digit and ones (units) digit of
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<math>\frac{n^2 + n + 15}{n}</math>
 
<math>\frac{n^2 + n + 15}{n}</math>
 
is an integer. Determine all possible values of <math>n</math>.
 
is an integer. Determine all possible values of <math>n</math>.
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==Solution==
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(a) Perform casework on the ones digit since it must be odd. If it is <math>1</math>, then the tens and hundreds digit will be the same, which is not permitted. If it is <math>3</math>, then <math>m=623</math> is valid. Otherwise, the tens digit will be <math>1</math>, so digits will not be distinct; thus <math>\boxed{m=623}</math>.
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(b) Since <math>1:4=20:80</math>, she has <math>20</math> and <math>80</math> black and gold marbles, respectively. Since <math>1:6=20:120</math>, she would need <math>120</math> gold marbles to create the new ratio, so she must add <math>120-80=\boxed{40}</math> gold marbles.
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(c) <math>\frac{n^2 + n + 15}{n}=n+1+\frac{15}{n}</math>, so all that is necessary is for <math>\frac{15}{n}</math> to be an integer. This is achieved when <math>n</math> is a positive factor of <math>15</math>; namely <math>\boxed{n=1,3,5,15}</math>.
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~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406]

Latest revision as of 21:15, 19 March 2025

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 $\boxed{n=1,3,5,15}$.

~ eevee9406

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