1969 AHSME Problems/Problem 24: Difference between revisions
Created page with "== Problem == When the natural numbers <math>P</math> and <math>P'</math>, with <math>P>P'</math>, are divided by the natural number <math>D</math>, the remainders are <math>R</..." |
Rockmanex3 (talk | contribs) Solution to Problem 24 |
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== Solution == | == Solution == | ||
== See | The divisors are the same, so take each variable [[modulo]] <math>D</math>. | ||
<cmath>P \equiv R \pmod{D}</cmath> | |||
<cmath>P' \equiv R’ \pmod{D}</cmath> | |||
That means | |||
<cmath>PP’ \equiv RR’ \pmod{D}</cmath> | |||
Thus, <math>PP’</math> and <math>RR’</math> have the same remainder when divided by <math>D</math>, so the answer is <math>\boxed{\textbf{(E)}}</math>. | |||
== See Also == | |||
{{AHSME 35p box|year=1969|num-b=23|num-a=25}} | {{AHSME 35p box|year=1969|num-b=23|num-a=25}} | ||
[[Category: Introductory Number Theory Problems]] | [[Category: Introductory Number Theory Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 19:06, 22 July 2018
Problem
When the natural numbers 
and 
, with 
, are divided by the natural number 
, the remainders are 
and 
, respectively. When 
and 
are divided by , the remainders are 
and 
, respectively. Then:
Solution
The divisors are the same, so take each variable modulo .
![\[P \equiv R \pmod{D}\]](http://latex.artofproblemsolving.com/3/b/8/3b8cdaceed647dae9e4e606601fd999465140125.png)
![\[P' \equiv R’ \pmod{D}\]](http://latex.artofproblemsolving.com/8/f/0/8f087839bc20d70399cc8044b8aff9acdd324d47.png)
That means
![\[PP’ \equiv RR’ \pmod{D}\]](http://latex.artofproblemsolving.com/2/b/0/2b0af997819124e7d515103839f03ef5a612c3f5.png)
Thus, 
and 
have the same remainder when divided by , so the answer is 
.
See Also
| 1969 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
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