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Cloudstone bank sat on the city’s busiest street. Its grand marble columns glimmered under the midday sun, and its copper roof made the whole structure glow. The entrance was guarded by two men wearing a badge in the shape of a shield: the symbol of this city’s police force.
Bernoulli entered the building itself. The reception was no less impressive, with a spacious interior, and a painted fresco on the ceiling. It was here that he found Hansel.
“Alright Constable, give me the scoop”
“Certainly”, replied Hansel. “At 1:03 this morning, the alarm in the vault was tripped. The police were immediately called, and arrived on scene at 1:11. However, there were no intruders to be found. In addition, there appeared to be no signs of breaking in, or anything taken out of the vault”
“Really?”, Bernoulli pondered. Was it really possible for the thieves to take nothing? “Let me see the vault”.
Hansel led the detective down a flight of stairs, and through heavy steel doors into a small room containing vast sums of money. Rows and rows of boxes, each containing piles of bills and coins. There was so much money that bills were strewn across the floor.
Bernoulli picked up one of the 100 euro bills on the floor. He stared at it for a while, and suddenly had an idea.
“These bills. Could they be counterfeit?”
“No. They come directly from the mint. They must be real.”
“Then, let’s test the property of these bills.” He pulled out a 100 euro bill from his wallet. “Did you know that the serial number of the euro bill has a specific property?”
“No, I wasn’t aware”
“All Euro bills have a serial number composed of a letter followed by eleven digits. Each letter is associated with a number from 1 to 9, such that if we replace the letter with the corresponding number, the final twelve digit number is a multiple of 9.
So for example, a bill could have the serial number Z00000000009. As it turns out, Z is associated with 9, and 900000000009 is divisible by 9, thus Z00000000009 is a valid serial number
I have a real Euro bill and a Euro bill from the vault, both with the same letter. We can get the value of the letter from the real Euro bill, and then check if the serial number of the other Euro bill satisfies our property”
“But”, Hansel interrupts, “How do we check if a number is a multiple of 9?”
“Remember, Constable, that we can take the digits of our number and add them together. If the resulting number is a multiple of 9, then our original number is a multiple of 9. For example, to check if 64251 is a multiple of 9, all we need to do is add its digits: 6+4+2+5+1 = 18. 18 is a multiple of 9. Thus, by our test, 64251 is a multiple of 9. If our test fails, then our original number is not a multiple of 9.”
With this information, can you find the value of L, and check if the vault bill has a valid serial number?
Real Euro Bill: L91562946386
Vault Euro Bill: L74385108965
Scroll down for the solution.
“Well, let’s start by adding all of the numbers together. 9+1+5+6+2+9+4+6+3+8+6 is of course… er… 10… 16… carry the 4… Constable, would you happen to have a calculator?”
After a bit of searching, a calculator from one of the clerk’s desk was found. It was found that 9+1+5+6+2+9+4+6+3+8+6 = 59.
“So, since the next multiple of 9 is 63, we get that L is 4.”
Now, adding the serial number on the new bill gets us 4 (our value for L) +7+4+3+8+5+1+0+8+9+6+5 = 60, which is not a multiple of 9. Thus, we know that this must be a counterfeit bill.
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