Brain Teasers
RomanKachi Locker !!!
I went to attend the funeral of my uncle, Randervaz the second, of the rich and famous Romankachi dynasty. He'd died the previous day after a long ailment of cancer. Having attended the funeral, I met my uncle's lawyer on the way out from the cemetery. He told me that my uncle had bequeathed part or all of his property to me in his will. However, he'd given strict instructions to his lawyer not to read this will until I satisfy a condition mentioned in his will. I had to get a legal document with my uncle's signature on it which said that now the will could be read out. However, this was locked away in one of the lockers of the Romankachi state bank. Saying this, he handed me an envelope addressed to me by my uncle. The envelope had a clue in it to lead me to the specified locker.
When I opened the letter, it read as follows :-
"Dearest nephew, you must already be knowing that you have to get a document with my signature on it from the Romankachi state bank in order for the will to be read out. All that I can tell you is that the locker is numbered 'bcb', such that
(ab)*(ba)=(bcb),
where a, b and c are all distinct positive integers from 1 to 9. The number 'ab' represents the two digit number whose 10's digit is a and units digit is b. The numbers 'ba' and 'bcb' also have similar meanings. Best of luck, Ruddy."
Can you find the locker number and help me out?
When I opened the letter, it read as follows :-
"Dearest nephew, you must already be knowing that you have to get a document with my signature on it from the Romankachi state bank in order for the will to be read out. All that I can tell you is that the locker is numbered 'bcb', such that
(ab)*(ba)=(bcb),
where a, b and c are all distinct positive integers from 1 to 9. The number 'ab' represents the two digit number whose 10's digit is a and units digit is b. The numbers 'ba' and 'bcb' also have similar meanings. Best of luck, Ruddy."
Can you find the locker number and help me out?
Answer
The equation given in the letter is(ab)*(ba)=bcb
Now (10a + b)*(10b + a) = bcb, given that the number 'ab' represents the two digit number whose 10's digit is a and units digit is b and so on for 'ba' and 'bcb'.
Hence 100*a*b + 10*(a^2 + b^2) + a*b = bcb, where ^ represents the operation "to the power of".
This is possible if ab = b and a^2 + b^2 = c, that is if a = 1 and hence 1 + b^2 = c.
Trying out various number from 1 to 9 for c in the above equation, I strike upon a combination that satisfies this equation 1 + b^2 = c, i.e. 1 + 2^2 = 5.
Thus a = 1, b = 2 and c = 5.
Therefore the locker number must be 252.
Verifying (ab)*(ba)=bcb, I got (12)*(21)=252
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