Difference between revisions of "1989 AIME Problems/Problem 3"

Revision as of 12:52, 30 August 2020

Problem

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if

$\frac{n}{810}=0.d25d25d25\ldots$

Solution

Repeating decimals represent rational numbers. To figure out which rational number, we sum an infinite geometric series, $0.d25d25d25\ldots = \sum_{i = 1}^\infty \frac{d25}{1000^n} = \frac{100d + 25}{999}$ . Thus $\frac{n}{810} = \frac{100d + 25}{999}$ so $n = 30\frac{100d + 25}{37} =750\frac{4d + 1}{37}$ . Since 750 and 37 are relatively prime, $4d + 1$ must be divisible by 37, and the only digit for which this is possible is $d = 9$ . Thus $4d + 1 = 37$ and $n = \boxed{750}$ .

(Note: Any repeating sequence of $n$ digits that looks like $0.a_1a_2a_3...a_{n-1}a_na_1a_2...$ can be written as $\frac{a_1a_2...a_n}{10^n-1}$ , where $a_1a_2...a_n$ represents an $n$ digit number.)

Solution 2

To get rid of repeating decimals, we multiply the equation by 1000. We get $\frac{1000n}{810} = d25.d25d25...$ We subtract the original equation from the second to get $\frac{999n}{810}=d25$ We simplify to $\frac{37n}{30} = d25$ Since $\frac{37n}{30}$ is an integer, $n=(30)(5)(2k+1)$ because $37$ is relatively prime to $30$ , and d25 is divisible by $5$ but not $10$ . The only odd number that yields a single digit $d$ and 25 at the end of the three digit number is $k=2$ , so the answer is $\boxed{750}$ .

Solution 3

Similar to Solution 2, we start off by writing that $\frac{1000n}{810} = d25.d25d25 \dots$ .Then we subtract this from the original equation to get:

$\frac{999n}{810} =d25 \Longrightarrow \frac{37n}{30} = d25 \Longrightarrow 37n = d25 \cdot 30$

Since n is an integer, we have that $37 \mid d25 \cdot 30$ .

Since $37$ is prime, we can apply Euclid's Lemma (which states that if $p$ is a prime and if $a$ and $b$ are integers and if $p \mid ab$ , then $p \mid a$ or $p \mid b$ ) to realize that $37 \mid d25$ , since $37 \nmid 30$ . Then we can expand $d25$ as $25 \cdot (4d +1)$ . Since $37 \nmid 25$ , by Euclid, we can arrive at $37 \mid 4d+1 \Longrightarrow d=9$ . From this we know that $n= 25 \cdot 30 = \boxed{750}$ . (This is true because $37n = 925 \cdot 30 \rightarrow n= 25 \cdot 30 = 750$ )

~qwertysri987

Solution 4

Write out the equations:

$\frac{n}{180} = \frac{d25}{999}$

$\frac{n}{30} = \frac{d25}{37}$

$37n = 30(d25)$

Thus $n$ divides 25 and 30. The only solution for this under 1000 is $\boxed{750}$ .

-jackshi2006

Revision as of 12:51, 30 August 2020 (view source) Jackshi2006 (talk \| contribs) (→‎Solution 4) ← Older edit		Revision as of 12:52, 30 August 2020 (view source) Jackshi2006 (talk \| contribs) (→‎Solution 4) Newer edit →
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−	Thus <math>n</math> divides 25 and 30. The ~~LCM~~ is <math>\boxed{750}</math>.	+	Thus <math>n</math> divides 25 and 30. The only solution for this under 1000 is <math>\boxed{750}</math>.

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