Question

If (x + 2y – 1) (x + 3y – 3) = 1271, where x and y are the natural numbers and (x + 2y  –  1), (x  +  3y  –  3) are prime numbers, then the value of $\frac{y}{x}$ is equal to

• A. $\frac{9}{10}$
• B. $\frac{3}{2}$
• C. $\frac{2}{3}$
• D. 1

Answer 1

The correct option is B $\frac{3}{2}$

Given: (x + 2y – 1) (x + 3y – 3) = 1271, where x and y are natural numbers. Also, it is given that (x + 2y – 1) and (x + 3y – 3) are both prime numbers.
Now, 1271 can be written as 31 × 41.
Since x and y are natural numbers,
∴ x + 3y – 3 > x + 2y – 1
Thus, x + 3y – 3 = 41 ..…(i)
and, x + 2y – 1 = 31 ..…(ii)
Subtracting (ii) from (i), we get
(x + 3y – 3) – (x + 2y – 1) = 41 – 31
⇒ x + 3y – 3 – x – 2y + 1 = 10
⇒ y – 2 = 10
⇒ y = 12
Putting y = 12 in (i), we get
x + (3 × 12) – 3 = 41
⇒ x + 36 – 3 = 41
⇒ x + 33 = 41
⇒ x = 8
Thus, x = 8 and y = 12.
$\frac{y}{x}=\frac{12}{8}=\frac{3}{2}$
Hence, the correct answer is option (2).