Number Theory

Correct Answer : Option d

Let the two-digit number be P = 10a + b, where a and b are the tens and unit’s digits respectively.
Given that the sum of the digits is 8, so a + b = 8.
Also, the number formed by interchanging the digits is 18 more than the original, so:
10b + a = P+18 = 10a + b + 18

Simplifying:
10b + a = 10a + b + 18
10b – b + a − 10a = 18

b – a = 2

Now solve the system:

1. a + b = 8

2. b – a = 2

So, a = 3 and b = 5

Therefore, the original number P = 10a + b = 35

Now check the statements:
Statement I is incorrect as 35 is not a prime number (divisible by 5 and 7)
Statement II: P + 14 = 35 + 14 = 49, which is divisible by 7 but not by 2,

So, Statement II is also incorrect.

Hence, the correct option is d.

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