A cheque was written for M rupees and N paisa, both M and N being-two digit numbers, but was mistakenly cashed for N rupees and M paisa. The incorrect amount exceeded the correct amount by Rs.1782, Then
A. M cannot be more than 70
B. N can be equal to 2M
C. The amount of the cheque cannot be multiple of 5.
D. The incorrect amount can be twice the correct amount.

Hint: In order to solve this problem we need to use the trial and error method. The trial and error method means that we will take the given option one by one and the one which will satisfy the given condition will be the correct answer, this whole process involves a lot of trials and errors hence called trial and error method.

Complete step by step solution:
We have given that,
The correct amount on the cheque is M rupees and N paisa,
It can be written as given below:
\[\Rightarrow 100M+N\] Paid
Given that the mistaken amount is N rupees and M paisa,
It can be written as given below:
\[\Rightarrow 100N+M\] Paid
Also given that the incorrect amount exceeded the correct amount by \[Rs.1782\].
It can be written as,
\[\Rightarrow 100N+M-\left( 100M+N \right)=1782\]
\[\Rightarrow 100N-N+M-100M=1782\]
On expanding the above expression and combining the like terms, we will get
\[\Rightarrow 99\left( N-M \right)=1782\]
\[\Rightarrow N-M=\dfrac{1782}{99}\]
\[\Rightarrow N-M=18\]
Here the above expression, if
\[N=89N=89\ and\text{ }M=71M=71\]
\[\therefore M=71,N=89\]
Here M is greater than 70.
Hence the first option is incorrect.
Now the correct amount on the cheque is \[100M+N\]
Let,
\[M=12M\Rightarrow M=12\ and\text{ }N=30N\Rightarrow N=30\]
Then the total amount is given by:
\[\Rightarrow 100\left( 12 \right)+30=1200+30=1230\]
Thus the correct amount is 1230 which is a multiple of 5.
Here the amount can be a multiple of 5.
Hence the third option is incorrect.
In the above expression we obtained that,
\[N-M=18\]
Now checking with the fourth option whether the incorrect amount can be twice the correct amount.
We have given that,
The correct amount is \[100M+N\] and the incorrect amount is \[100N+M\],
Expressing the fourth option mathematically as;
\[\Rightarrow 100N+M=2\left( 100M+N \right)\]
\[\Rightarrow 100N+M=200M+2N\]
\[\Rightarrow 98N=198M\]------ (1)
\[\Rightarrow N-M=18\]------ (2)
In solving these equations we get the solutions of M and N as non-integers.
As we know that,
But given that the M and N are two-digit numbers which are integers. Hence the incorrect amount cannot be twice the correct amount.
Thus,
The fourth option is incorrect.
Now as given M and N can be any two digit numbers,
So,
Let \[M=18M\Rightarrow M=18\ and\text{ }N=36N\Rightarrow N=36\]
Here,
We can observe that,
\[N=2M\]
Therefore,
N can be equal to 2M.

Hence, the option (B) is the correct answer.

Note: Students need to know that the trial and error method deals with the assumptions and if errors come in it, then we go to another assumption and substitute till we do not get any errors. Here while solving this problem, as in the most of the involved process we are dealing with the trial and error method, we substituted all the given hints and clues and then started doing the trial and error method, here while finding whether N can be equal to 2M, then here M and N can be any two-digit numbers for example we can take\[M=12\text{ }and\text{ }N=24\], where the statement “N can be equal to 2M” still holds true.