Difference between revisions of "1969 AHSME Problems/Problem 24"
(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 20:06, 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 . That means 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 | |
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All AHSME Problems and Solutions |
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