Difference between revisions of "1999 AHSME Problems/Problem 25"
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~aopspandy | ~aopspandy | ||
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+ | == Solution 3 (The Easiest and Most Intuitive) == | ||
+ | Let's clear up the fractions: <cmath>\frac{5}{7}=\frac{2520a_2+840a_3+210a_4+42a_5+7a_6+a_7}{7!}</cmath> | ||
+ | <cmath>3600=2520a_2+840a_3+210a_4+42a_5+7a_6+a_7</cmath> | ||
+ | Notice that if we divide everything by <math>7</math> then we would have:<cmath>514+\frac{2}{7}=360a_2+120a_3+30a_4+6a_5+a_6+\frac{1}{7}a_7</cmath> | ||
+ | Since <math>0 \le a_7<7</math> and <math>a_7</math> must be an integer, then we have <math>\frac{2}{7}=\frac{1}{7}a_7</math>, so <math>a_7=2</math>. | ||
+ | |||
+ | Similarly, if we divide everything by <math>6</math>, then we would have: <cmath>85+\frac{4}{6}=60a_2+20a_3+5a_4+a_5+\frac{1}{6}a_6</cmath> | ||
+ | Again, since <math>0 \le a_6<6</math> and <math>a_6</math> must be an integer, we have <math>\frac{4}{6}=\frac{1}{6}a_6</math>, so <math>a_6=4</math>. | ||
+ | |||
+ | The pattern repeats itself, so in the end we have <math>a_2=1</math>, <math>a_3=1</math>, <math>a_4=1</math>, <math>a_5=0</math>, <math>a_6=4</math>, <math>a_7=2</math>. So <math>a_2+a_3+a_4+a_5+a_6+a_7=\boxed{\textbf{(B)~9}}</math> | ||
+ | |||
+ | ~BurpSuite, with a help from ostriches88 | ||
+ | |||
+ | ==Solution 4== | ||
+ | |||
+ | By multiplying both sides by <math>7</math> we get | ||
+ | |||
+ | <cmath>5 = \frac72 a_2 + \frac76 a_3+ \frac{7}{24} a_4 + \frac{7}{120} a_5 + \frac{7}{720} a_6 + \frac{a_7}{720}</cmath> | ||
+ | |||
+ | since <math>0 \le a_2 < 2</math>, if <math>a_2 = 0</math> the rest of the right hand side will not add up to be <math>5</math>, so <math>a_2 = 1</math> | ||
+ | |||
+ | <cmath>\frac32 = \frac76 a_3+ \frac{7}{24} a_4 + \frac{7}{120} a_5 + \frac{7}{720} a_6 + \frac{a_7}{720}</cmath> | ||
+ | |||
+ | If <math>a_3 = 2</math>, <math>\frac76 a_3 = \frac73 > \frac32</math>, so <math>a_3 = 1</math> | ||
+ | |||
+ | <cmath>\frac13 = \frac{7}{24} a_4 + \frac{7}{120} a_5 + \frac{7}{720} a_6 + \frac{a_7}{720}</cmath> | ||
+ | |||
+ | If <math>a_4 = 2</math>, <math>\frac{7}{24} a_4 = \frac{7}{12} > \frac13</math>, so <math>a_4 = 1</math> | ||
+ | |||
+ | <cmath>\frac{1}{24} = \frac{7}{120} a_5 + \frac{7}{720} a_6 + \frac{a_7}{720}</cmath> | ||
+ | |||
+ | If <math>a_5 = 1</math>, <math>\frac{7}{120} a_5 = \frac{7}{120} > \frac{1}{24}</math>, so <math>a_5 = 0</math> | ||
+ | |||
+ | <cmath>\frac{1}{24} = \frac{7}{720} a_6 + \frac{a_7}{720}</cmath> | ||
+ | |||
+ | <math>30 = 7 a_6 + a_7</math>. Since <math>a_7 < 7</math>, <math>a_7 = 2</math> and <math>a_6=4</math> | ||
+ | |||
+ | Therefore, <math>a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7} = 1 + 1 + 1 + 0 + 4 + 2= \boxed{\textbf{(B) } 9}</math>. | ||
+ | |||
+ | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
+ | |||
== See also == | == See also == | ||
{{AHSME box|year=1999|num-b=24|num-a=26}} | {{AHSME box|year=1999|num-b=24|num-a=26}} |
Latest revision as of 01:43, 4 October 2023
Contents
Problem
There are unique integers such that
where for
. Find
.
Solution 1(Modular Functions)
Multiply out the to get
By Wilson's Theorem (or by straightforward division), , so
. Then we move
to the left and divide through by
to obtain
We then repeat this procedure , from which it follows that
, and so forth. Continuing, we find the unique solution to be
(uniqueness is assured by the Division Theorem). The answer is
.
Solution 2(Basic Algebra and Bashing)
We start by multiplying both sides by , and we get:
After doing some guess and check, we find that the answer is
.
~aopspandy
Solution 3 (The Easiest and Most Intuitive)
Let's clear up the fractions:
Notice that if we divide everything by
then we would have:
Since
and
must be an integer, then we have
, so
.
Similarly, if we divide everything by , then we would have:
Again, since
and
must be an integer, we have
, so
.
The pattern repeats itself, so in the end we have ,
,
,
,
,
. So
~BurpSuite, with a help from ostriches88
Solution 4
By multiplying both sides by we get
since , if
the rest of the right hand side will not add up to be
, so
If ,
, so
If ,
, so
If ,
, so
. Since
,
and
Therefore, .
See also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Problem 26 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.