Question

It is given that the number $43361$ can be written as a product of two distinct prime numbers $p_1,p_2$. Further, assume that there are $42900$ numbers which are less than $43361$ and are co-prime to it. Then, $p_1+p_2$ is

• A. $462$
• B. $464$
• C. $400$
• D. $402$

Answer 1

Answer: (A)

$462$

Let's take $n=43361$ and given that $n=p_1.p_2$ where $p_1$, $p_2$ are two prime numbers.

and also given that there are $42900$ numbers which are less than $43361$ and are co-prime to it. Therefore from Euler's totient function $\varphi$ which states as number of numbers which are less than $n$ and co-prime to it and it is given by $\varphi \left(n\right)=n-1$ if $n$ is prime else $\varphi \left(n\right)=\varphi \left(p_1\right).\varphi \left(p_2\right)$ if $n$ is written as product of those two numbers.

Now, $\varphi \left(43361\right)=42900=(p_1-1)(p_2-1)$ $⟹$ $42900=43361-(p_1+p_2)+1$ $⟹$ $p_1+p_2=462$