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## What is Parity of a number?

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.

## Try the problem

In how many ways can $10001$ be written as the sum of two primes?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$.

Source
Competency
Difficulty
Suggested Book

AMC 8, 2011 Problem number 24

Parity of a number and properties of prime numbers.

5 out of 10

Mathematical Circles

## Use some hints

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.