Difference between revisions of "1966 AHSME Problems/Problem 39"
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== See also == | == See also == | ||
| − | {{AHSME box|year=1966|num-b=38|num-a=40}} | + | {{AHSME 40p box|year=1966|num-b=38|num-a=40}} |
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 11:02, 29 July 2024
Problem
In base 
the expanded fraction 
becomes 
, and the expanded fraction 
becomes 
. In base 
fraction , when expanded, becomes 
, while the fraction becomes 
. The sum of and , each written in the base ten, is:
Solution
First, let's write as a proper fraction in base . To do that, note that:
Multiplying this equation on both sides 
, we get:
Subtracting the first equation from the second one, we get:
Using a very similar method as above, we can see that:
and 
.
Dividing the 2 equations to get some potential solutions, we get (note that we don't have to worry about division by zero because 
):
Since non-integer bases are rarely used, we can try to assume that the solution is positive integers and see if we get any solutions.
Notice that 
has to be a negative factor of 840. We need to plug in values of 
. 840 divides -21, so we plug in 8 to check. Luckily, when 
, we see that 
, and furthermore, 11+8=19 is one of the answers. We can quickly test it into the original equations and , and we see they indeed work. Therefore the answer is 
See also
| 1966 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 38 |
Followed by Problem 40 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
