Q 1. Vivek found the product, P, of two two-digit natural numbers, M
and N. He then reversed the digits of each of Mand N and found the
product of the resultant numbers. Interestingly, he found both
products to be the same. If the product of the tens digit of M and the
tens digit of N is prime, find the sum of all the possible values of P
that Vivek could have obtained.
(1) 2604
(2) 2712
(3) 2627
(4) 4684
(5) 4664
Answers
4684 is the sum of all the possible values of P that Vivek could have obtained
Step-by-step explanation:
Let say M = AB
& N = CD
P = M*N
P =(10A + B)(10C + D)
=> P = 100*A*C + 10A*D + 10B*C + B*D
Digit reverse
P = (10B + A)(10D + C)
=> P = 100B*D + 10B*C + 10A*D + A*C
100A*C + 10A*D + 10B*C + B*D = 100B*D + 10B*C + 10A*D + A*C
=> 99A*C = 99B*D
=> A*C = B*D
product of the tens digit of M and the tens digit of N is prime
=> A*C is Prime
A * C is Prime
1 * 2 , 1 * 3 , 1 * 5 , 1 * 7 ,
2 * 1 , 3 * 1 , 5 * 1 , 7 * 1
AB = 11 CD = 22 P = 242
AB = 12 CD = 21 P = 252
Below two cases will produce same P
AB = 21 CD = 12
AB = 22 CD = 11
AB = 13 CD = 31 P = 403
AB = 11 CD = 33 P = 363
AB = 15 CD = 51 P = 765
AB = 11 CD = 55 P = 605
AB = 17 CD = 71 P = 1207
AB = 11 CD = 77 P = 847
Adding all = 242 + 252 + 403 + 363 + 765 + 605 + 1207 + 847
= 4684