Question 85

Read the given statements and select the most appropriate option Statement-I: If the sum of remainders obtained when 3864335 divisible by 382 and 300 is twice the remainder when a least four-digit number is divisible by 32, then that number is 1021 Statement-II: 45!-1 is a prime number.

Solution

Statement 1:Â
Remainder of 3864335 when divided by 300 is a short calculation and gives us the remainder as 35.Â
Remainder of 3864335 when divided by 382 is lengthy but after dividing the remainder would be 23.
Sum of these remainders is 58, this is twice of the remainder we get on dividing the unknown number by 32;Â
This remainder would be 29
We now have to check the remainder when 1021 is divided by 32,Â
1024 isÂ $$32^2$$, soÂ $$\left[\frac{1024}{32}\right]_R=-3=\ 29$$
And hence, we can confirm that statement 1 is correct.Â

Statement 2:Â According to Wilson's theorem,Â
$$\left[\frac{\left(p-2\right)!}{p}\right]_R=1$$ where P is a prime number.Â
Taking P =47 we get,Â $$\left[\frac{45!}{47}\right]_R=1$$ and subtracting 1 from this would mean that 45!-1 is divisible by 47.Â
Therefore, Statement II is incorrect.Â