Question

If the sum of three prime numbers is 100 and one of them exceeds others by 36, then find the largest number.

Answer 1

Hint: First, try to understand the concept of a prime number. A prime number is one which has only two factors 1 & the number itself. Then, write down the equations supporting the given information & find out all prime numbers and then choose the largest prime number.

Complete step by step solution: First, we will understand what exactly is meant by prime numbers So, a prime number is that number which has only two factors 1 and the number itself ${\text{eg:}}\;{\text{2, 3, 5, 7, etc}}{\text{.}}$
Let the tree prime numbers be \[{\text{x }},\;{\text{y}}\;\text{and}\;{\text{z}}\]

It is given that sum of the three prime numbers is 100
\[i.e\;\;\;x + y + z = 100\;\] $\left( 1 \right)$
It is also given that one of the prime numbers exceeds another by 36. So, let y exceeds \[z\;\text{by}\;36\]

\[ \Rightarrow y = z + 36\;\;\;\;\] -eq (2)

Substituting \[eq \to \left( 2 \right)\] in equation \[ \to \left( 1 \right)\] we get
$Z + \left( {Z + 36} \right) + Z = 100$
\[ \Rightarrow \;x + {\text{2z}}\,{\text{ + 36 = 100}}\]
$ \Rightarrow \;x + 2z = 64$
We got,
$x + 2z = 64\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;2z = 64 - x$
Since, \[x\& z\] are both prime numbers and also as we have;
$2z = 64 - x.$
We must have \[x\] as some even prime numbers because if x is not even (say 5)
Then $2z = 64 - 5$
$2z = 59$
          Then $z = \dfrac{{59}}{2}$
Which is not a prime numbers as prime numbers are always among natural numbers
Therefore, we must have \[x\] as even prime numbers. There is only even a prime number ${\text{i}}{\text{.e}}\; \to {\text{2}}$.

\[\therefore \boxed{x = 2}\]
\[2z = 64 - 2\]
Substituting\[\;x = 2\] in above equation,
$2z = 64 - 2$
$ \Rightarrow \;2z = 62$
$ \Rightarrow \;z = \dfrac{{62}}{2} - 31\;\;\;\;\;\;\;\;\;\;\;\;\;\;\therefore \boxed{z = 31}$
From equation (2), we have

$y = z + 36$
$ = 31 + 36$
$\boxed{y = 67}$
$\therefore \;x = 2,y = 67,z = 31$
\[\therefore \] Among,\[\;x,{\text{ }}y\& z;\] largest prime number is y whose value is 67.
\[\therefore \] Largest prime number \[ = 67.\]

Note: Here, we had to choose \[x\] by trial & error method only as we can only have \[x\] as even prime number. Also, you must remember that 2 is the only even prime number.