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Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 11"

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==Solution 4 (Very algebraic)==
 
==Solution 4 (Very algebraic)==
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If <math>a</math> divides <math>252,</math> then <math>3a</math> and <math>4a</math> must also divide <math>252.</math> This implies that <math>\frac{252}{3a},\frac{252}{4a}</math> are both integers, and that <math>a</math> divides and multiplying, we have that <math>a</math> divides <math>84</math> and <math>63.</math> The greatest common divisor of <math>84</math> and <math>63</math> is <math>21,</math> and we can check that indeed <math>a=\boxed{21}.</math>
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If <math>a</math> divides <math>252,</math> then <math>3a</math> and <math>4a</math> must also divide <math>252.</math> This implies that <math>\frac{252}{3a},\frac{252}{4a}</math> are both integers, and that <math>a</math> divides and multiplying, we have that <math>a</math> divides <math>84</math> and <math>63.</math> The greatest common divisor of <math>84</math> and <math>63</math> is <math>21,</math> and we can check that indeed <math>a=\boxed{21}.</math> ~samrocksnature
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==See also==
 +
#[[2021 JMPSC Accuracy Problems|Other 2021 JMPSC Accuracy Problems]]
 +
#[[2021 JMPSC Accuracy Answer Key|2021 JMPSC Accuracy Answer Key]]
 +
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
 +
{{JMPSC Notice}}

Latest revision as of 00:17, 12 July 2021

Problem

If $a : b : c : d=1 : 2 : 3 : 4$ and $a$, $b$, $c$, and $d$ are divisors of $252$, what is the maximum value of $a$?

Solution 1

$a$ must be a number such that $2a \mid 252$, $3a \mid 252$, $4a \mid 252$. Thus, we must have $12a \mid 252$. This implies the maximum value of $a$ is $252/12 = \boxed{21}$, which works.

~Bradygho

Solution 2

Notice that $252=2^2\cdot 3^2\cdot 7$. Because $b=2a$ and $d=4a,$ it is invalid for $a$ to be a multiple of $2$. With similar reasoning, $a$ must have at most one factor of $3$. Thus, $a=\boxed{21}$.


(With $a=21$, we have $b=42, c=63, d=84,$ which is valid)

~Apple321

Solution 3 (A Little Bashy)

Note $252=2^2 \cdot 3^2 \cdot 7$, so the divisors are $\{1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252 \}$. We see the set $\{21,42,63,84 \}$ is the largest 4-digit set we can form, so the answer is $a=\boxed{21}$ $\linebreak$ ~Geometry285

Solution 4 (Very algebraic)

If $a$ divides $252,$ then $3a$ and $4a$ must also divide $252.$ This implies that $\frac{252}{3a},\frac{252}{4a}$ are both integers, and that $a$ divides and multiplying, we have that $a$ divides $84$ and $63.$ The greatest common divisor of $84$ and $63$ is $21,$ and we can check that indeed $a=\boxed{21}.$ ~samrocksnature

See also

  1. Other 2021 JMPSC Accuracy Problems
  2. 2021 JMPSC Accuracy Answer Key
  3. All JMPSC Problems and Solutions

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