Difference between revisions of "Parity"
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'''Parity''' refers to whether a number is [[even]] or [[odd]]. | '''Parity''' refers to whether a number is [[even]] or [[odd]]. | ||
| − | While this may seem highly basic, checking the parity of numbers is often an useful tactic for solving problems, especially with [[proof by contradiction]]s and [[casework]]. | + | While this may seem highly basic, checking the parity of numbers is often an useful tactic for solving problems, especially with [[proof by contradiction]]s and [[casework]]. |
| + | |||
| + | This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1. | ||
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| + | This only applies to integers, not fractions or decimals. | ||
| + | |||
| + | == Problems == | ||
=== Introductory === | === Introductory === | ||
*Find <math>\gcd(a+b,a^2+b^2)</math> if <math>\gcd(a,b) = 1</math>. | *Find <math>\gcd(a+b,a^2+b^2)</math> if <math>\gcd(a,b) = 1</math>. | ||
Revision as of 19:16, 28 January 2016
Parity refers to whether a number is even or odd.
While this may seem highly basic, checking the parity of numbers is often an useful tactic for solving problems, especially with proof by contradictions and casework.
This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1.
This only applies to integers, not fractions or decimals.
Problems
Introductory
- Find
if 
.
if 
.Intermediate
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