Question

 The number which is four more than the square of 625 has exactly two prime factors. Determine what they are?
A. 577, 677
B. 517, 617
C. 437, 537
D. 411, 523

Answer 1

Hint: First we will find the number from the statement given in the question. So, we will get the number as \[{{\left( 625 \right)}^{2}}+4\]. And then we will find the prime factors of that number using the prime factorization technique.

Complete step-by-step answer:

First, we will find the number.
The number is four more than the square of 625. That means, the number is,
$\begin{align}
  & ={{\left( 625 \right)}^{2}}+4 \\
 & =390,625+4 \\
 & =390,629 \\
\end{align}$
Now, we have to find the prime factor of the number 390629.
But for that, first, we have to find what exactly the prime factors of a number are and before that, we have to know what a prime number is.
A prime number is a number which is divisible by itself and 1.
For example: 2, 3, 5, 7, 11.
A prime factor of a number is a factor of number and also it is a prime number. For example: The prime factor of 15 is 3 & 5 as $3\times 5=15$.
So, the prime factors of 390629 can be found as,
$\begin{align}
  & 577\left| \!{\underline {\,
  390,629 \,}} \right. \\
 & 677\left| \!{\underline {\,
  677 \,}} \right. \\
 & \ \ \ \ \ \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
The prime factors of 390629 are 577 and 677.

Note: Alternate Method
First, we will find the number exactly the way we find in the original method.
But as we know that prime factors are factors of a number. So, here we can use the options to find the right answer.
Option (A) 577, 677
As we can already see that both are prime numbers as the numbers which can divide these numbers are only themselves and 1. And by multiplying both numbers we get,
$577\times 677=390629$