August 2023
Problem of the Month
Whose sum is correct?
Submitted by K. Sengupta
Ned asked each of his 4 children to think of a 4-digit number.
"Now transfer the last digit to the front and add the new number to the old one.
For example, 1234 + 4123 = 5357.
Now tell me the results."
The results were told by each of the four children as follows:
Barry: 2348
Mary: 7847
Jaypee: 11847
Darla: 9846
"Only one of you gave me the correct sum" Ned told the gathering.
Who was it and how did Ned know?
Solution:
Only Jaypee's response is correct.
The sum of two such numbers is always divisible by 11.
Let the number be ABCD = 1000A + 100B + 10C + D
Then, placing the last digit to the front we have the number DABC = 1000D +100A + 10B + C
Then, ABCD + DABC = 1001D + 110A + 110B + 11C = 11(91D + 10A + 10B + C)
Now, Barry's response = 2348, which is NOT divisible by 11
Mary's response = 7847, which is NOT divisible by 11
Jaypee's response = 11847 = 11 * 1077
Darla's reponse = 9846, which is NOT divisible by 11
Hence, only Jaypee's response is accurate.