Difference between revisions of "1964 AHSME Problems/Problem 3"
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<math>x \equiv v\ (\textrm{mod}\ y)</math> <math>=></math> <math>x+2uy \equiv v\ (\textrm{mod}\ y)</math> | <math>x \equiv v\ (\textrm{mod}\ y)</math> <math>=></math> <math>x+2uy \equiv v\ (\textrm{mod}\ y)</math> | ||
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==Solution 2== | ==Solution 2== | ||
Latest revision as of 05:04, 31 December 2023
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
