Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?

396

Age 14 to 16 Challenge Level:

The four digits 5, 6, 7 and 8 are put at random in the spaces of
the number below:

3 _ 1 _ 4 _ 0 _ 9 2

Estimate the probability that the answer will be a multiple of
396.

Calculate the probability and explain your answer.