1964 AHSME Problems/Problem 3
Problem
When a positive integer 
is divided by a positive integer 
, the quotient is 
and the remainder is 
, where and are integers.
What is the remainder when 
is divided by ?
Solution 1
- We can solve this problem by elemetary modular arthmetic,
\equiv. (
) 
+ 
\equiv. ().
% Solution by GEOMETRY-WIZARD$==Solution 2==
By the definition of quotient and remainder, problem states that$ (Error compiling LaTeX. Unknown error_msg)x = uy + v$.
The problem asks to find the remainder of$ (Error compiling LaTeX. Unknown error_msg)x + 2uy
y
2uy
y
x
\boxed{\textbf{(D)}}$.
==Solution 3==
If the statement is true for all values of$ (Error compiling LaTeX. Unknown error_msg)(x, y, u, v)
(x, y, u, v)$.
If you let$ (Error compiling LaTeX. Unknown error_msg)x=43
y = 8
u = 5
v = 3
x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123
8
3$.
When you plug in$ (Error compiling LaTeX. Unknown error_msg)u=5v = 3
0, 5, 10, 3, 6
\boxed{\textbf{(D)}}$.
See Also
| 1964 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
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| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
