If x and y are two-digit prime numbers such that y is obtained from x by interchanging its digits and x – y = 36, then what is the value of xy ?
[amp_mcq option1=”1611″ option2=”2701″ option3=”4031″ option4=”5603″ correct=”option2″]
The number y is obtained by interchanging the digits, so y = 10b + a.
Both x and y must be prime numbers.
We are given that x – y = 36.
(10a + b) – (10b + a) = 36
9a – 9b = 36
9(a – b) = 36
a – b = 4.
We need to find two digits a and b (1-9) such that a – b = 4, and both (10a + b) and (10b + a) are prime numbers.
Let’s list the possible pairs (a, b) where a – b = 4:
– If b = 1, a = 5. x = 51 (51 = 3 * 17, not prime).
– If b = 2, a = 6. x = 62 (not prime).
– If b = 3, a = 7. x = 73. 73 is a prime number. y = 37. 37 is a prime number. This pair (a=7, b=3) satisfies all conditions.
– If b = 4, a = 8. x = 84 (not prime).
– If b = 5, a = 9. x = 95 (not prime).
The only pair of digits satisfying the conditions is a=7 and b=3.
So, x = 73 and y = 37.
Check: x and y are two-digit prime numbers. x – y = 73 – 37 = 36. All conditions met.
The question asks for the value of xy.
xy = 73 * 37.
Calculation:
73 * 37 = 73 * (30 + 7) = 73 * 30 + 73 * 7
= 2190 + 511
= 2701.