Sum of the two-digit number is 8 and when digits of the two-digit number are interchanged the new number is 18 more than the original. If original number is P, then consider the following statement:

Statement I: P is a prime number.

Statement II: P + 14 = Q2 and Q is divisible by 2 and 7.

Which of the following statement is correct?

(a) Only Statement I

(b) Only Statement II

(c) Both Statement I and II

(d) Neither Statement I nor II

Question

Sum of the two-digit number is 8 and when digits of the two-digit number are interchanged the new number is 18 more than the original. If original number is P, then consider the following statement: 

Statement I: P is a prime number. 

Statement II: P + 14 = Q2 and Q is divisible by 2 and 7. 

Which of the following statement is correct? 

(a) Only Statement I 

(b) Only Statement II

(c) Both Statement I and II 

(d) Neither Statement I nor II

ONLiNE UPSC · Accepted Answer

Let the two-digit number be P = 10a + b, where a and b are the tens and unit&rsquo;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 &ndash; b + a &minus; 10a = 18

b &ndash; a = 2

Now solve the system:

1. a + b = 8

2. b &ndash; 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.

Number Theory

Two-digit numbers

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