Parity is a property of a number tells that if it is even or odd. If a number is odd then it is said to be of odd parity and if it is even then it is said to be of even parity.
In how many ways can $10001$ be written as the sum of two primes?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$.
AMC 8, 2011 Problem number 24
Parity of a number and properties of prime numbers.
5 out of 10
Mathematical Circles
First hint
$10001$ is a number of $\textbf{ODD PARITY}$.
Is it possible to get an odd number by adding two numbers of same parity ? What do you think ?
Second Hint
Answer to the above question is : NO !
If $2m+1$ and $2n+1$ are two odd numbers then by adding we get $2(m+n+1)$, which is even.
Similarly $2m+2n=2(m+n)$
Then the two numbers must be of different parity to get a odd number by adding them.
$[2m+2n+1=2(m+n)+1]$
Third Hint
If $A+B=10001$ where $A$ and $B$ are primes.
then one of $A$ or $B$ must be even.
Final Step
But the only even prime number is $2$.
i.e., either $A$ or $B$ is equal to $2$
Let $A=2$ then $B=10001-2=9999$
But $9999$ is not a prime number since it is divisible by $3$.
Then there are $\textbf{NO}$ such ways to write $10001$ as a sum of two prime numbers.
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