Difference between revisions of "1964 AHSME Problems/Problem 3"
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| − | We can | + | ==Problem== |
| + | |||
| + | 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== | ||
| + | |||
| + | *We can solve this problem by elemetary modular arthmetic, | ||
| + | <math>x \equiv v\ (\textrm{mod}\ y)</math> <math>=></math> <math>x+2uy \equiv v\ (\textrm{mod}\ y)</math> | ||
| + | |||
| + | ~GEOMETRY-WIZARD | ||
| + | |||
| + | ==Solution 2== | ||
| + | |||
| + | 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 3== | ||
| + | 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 16:19, 14 August 2025
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 | |
| 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 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
