AMC 8 USA Math Olympiad

Parity : AMC 8, 2011 PROBLEM 24

This is an a beautiful problem from AMC 8, 2011. Involves the concept of parity of a number. We provide sequential hint.

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?

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

AMC 8, 2011 Problem number 24

Parity of a number and properties of prime numbers.

5 out of 10

Mathematical Circles

Knowledge Graph

Use some hints

$10001$ 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 ?

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.


If $A+B=10001$ where $A$ and $B$ are primes.

then one of $A$ or $B$ must be even.

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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