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,
~GEOMETRY-WIZARD
Solution 2
By the definition of quotient and remainder, problem states that .
The problem asks to find the remainder of when divided by . Since is divisible by , adding it to will not change the remainder. Therefore, the answer is .
Solution 3
If the statement is true for all values of , then it must be true for a specific set of .
If you let and , then the quotient is and the remainder is . The problem asks what the remainder is when you divide by . In this case, the remainder is .
When you plug in and into the answer choices, they become , respectively. Therefore, the answer is .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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