Final answer:
Barry's 6-digit passcode, which forms a correct calculation when a multiplication sign is placed between the second and third digits and an equal sign between the third and fourth, considering the given digits, is found to be 615052.
Step-by-step explanation:
The student is asked to determine Barry's 6-digit passcode for Internet banking, given that the passcode forms a correct mathematical calculation when interpreted with a multiplication sign and an equal sign at specified positions. We know the passcode has six different digits, with the first digit being 6, the third digit being 5, and the fifth digit being 2. The second digit (when multiplied by the third digit, 5) should give a product that is a two-digit number, starting with 5, which limits our options for the second digit. The fourth and sixth digits should be a quotient of the two-digit number and the second digit, creating a valid mathematical operation.
To find the passcode, we begin by identifying the possible values for the second digit that, when multiplied by 5, result in a two-digit number starting with 5. The only digit that satisfies this condition is 1 (because 1 * 5 = 5). Thus, the multiplication gives us 5 as the fourth digit (since 5 * 1 = 05, and the multiplication equals the two digits following in the passcode). The remaining available digits for the sixth position would be 0, 3, 4, 7, 8, and 9. Among these, 0 cannot be chosen as it is neither a multiplication product of 10 nor a division result in our case. Therefore, we consider the remaining digits and find that 3, when multiplied by 2 (fifth digit), gives us 6, satisfying our equation (since 1 * 5 = 05 and 06/3 = 2). Hence, the complete passcode is 615052.