Difference between revisions of "1964 AHSME Problems/Problem 3"
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− | + | ==Problem== | |
− | Solution by | + | When a positive integer <math>x</math> is divided by a positive integer <math>y</math>, the quotient is <math>u</math> and the remainder is <math>v</math>, where <math>u</math> and <math>v</math> are integers. |
+ | What is the remainder when <math>x+2uy</math> is divided by <math>y</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ 2u \qquad | ||
+ | \textbf{(C)}\ 3u \qquad | ||
+ | \textbf{(D)}\ v \qquad | ||
+ | \textbf{(E)}\ 2v </math> | ||
+ | |||
+ | |||
+ | ==Solution 1== | ||
+ | |||
+ | By the definition of quotient and remainder, problem states that <math>x = uy + v</math>. | ||
+ | |||
+ | The problem asks to find the remainder of <math>x + 2uy</math> when divided by <math>y</math>. Since <math>2uy</math> is divisible by <math>y</math>, adding it to <math>x</math> will not change the remainder. Therefore, the answer is <math>\boxed{\textbf{(D)}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | If the statement is true for all values of <math>(x, y, u, v)</math>, then it must be true for a specific set of <math>(x, y, u, v)</math>. | ||
+ | |||
+ | If you let <math>x=43</math> and <math>y = 8</math>, then the quotient is <math>u = 5</math> and the remainder is <math>v = 3</math>. The problem asks what the remainder is when you divide <math>x + 2uy = 43 + 2 \cdot 5 \cdot 8 = 123</math> by <math>8</math>. In this case, the remainder is <math>3</math>. | ||
+ | |||
+ | When you plug in <math>u=5</math> and <math>v = 3</math> into the answer choices, they become <math>0, 5, 10, 3, 6</math>, respectively. Therefore, the answer is <math>\boxed{\textbf{(D)}}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME 40p box|year=1964|num-b=2|num-a=4}} | ||
+ | |||
+ | [[Category:Introductory Number Theory Problems]] | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 02:30, 23 July 2019
Contents
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
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 2
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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All AHSME Problems and Solutions |
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