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N is a multiple of 72. Also, it is known that all the digits of N are different. What is the largest possible value of N?
    Correct answer is '9876543120'. Can you explain this answer?
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    N is a multiple of 72. Also, it is known that all the digits of N are ...
    We know that the number is a multiple of 72. Therefore, the number must be a multiple of both 8 and 9. 
    If the number contains all the digits from 0 to 9, the sum of the digits will be 9*10/2 = 45.
    Therefore, a number containing all the ten digits will be divisible by 9.
    For a number to be divisible by 8, the last 3 digits should be divisible by 8.
    Now, we have to find the largest possible number. Therefore, the left-most digit must be 9, the second digit from the left should be 8 and so on. 
    We'll get 9876543210 as the number. However, the last 3 digits are not divisible by 8. We must try to make the last 3 digits divisible by 8 without altering the position of other numbers. '120' is divisible by 8.
    Therefore, the largest number with different digits divisible by 72 is 9876543120.
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    N is a multiple of 72. Also, it is known that all the digits of N are ...
    Question: What is the largest possible value of N if N is a multiple of 72 and all the digits of N are different?

    Approach:
    To find the largest possible value of N, we need to determine the arrangement of the digits such that N is divisible by 72.

    Divisibility Rule of 72:
    A number is divisible by 72 if it is divisible by both 8 and 9.

    Divisibility Rule of 8:
    A number is divisible by 8 if the last three digits of the number form a multiple of 8.

    Divisibility Rule of 9:
    A number is divisible by 9 if the sum of its digits is divisible by 9.

    Finding the Largest Possible Value of N:
    1. We start by considering the divisibility rule of 8. Since the last three digits of N must form a multiple of 8, we need to find a combination of digits that satisfies this condition.

    2. We can start with the largest digit, i.e., 9, and check if the remaining digits can form a multiple of 8.

    3. To form a multiple of 8, the last three digits of N can be 912, 928, 936, 952, 968, or 984.

    4. Next, we consider the divisibility rule of 9. Since all the digits of N must be different, the sum of the digits must be a multiple of 9.

    5. The sum of the digits from 1 to 9 is 45, which is divisible by 9. However, if we include 0 in the sum, the sum becomes 45 + 0 = 45, which is not divisible by 9. Therefore, we cannot include 0 in N.

    6. We can arrange the remaining digits (1 to 9) in descending order to form the largest possible value of N.

    Final Answer:
    The largest possible value of N is 9876543120.
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    Darren Puppet, the richest person in the world, has ensured extra security for his files. He implements a security based on division algorithms in his passwords. The passwords contain n digits such that :I. It contains all digits from 1 to n, and all of them are used exactly once.II. The number formed by the first two digits is divisible by 2, the number formed by the first 3 digits is divisible by 3, and so on such that the n digit password is divisible by n. (For example, 123 is such a 3 digit password, 12 is divisible by 2 and 123 by 3).To remember which password he is using, Darren ranks all the passwords having certain number of digits in increasing order and hence by just knowing the number of digits and the rank, he is able to correctly enter the password. (For example, for 3 digits passwords, 123 has rank 1 and 321 has rank 2)It is known that he uses 6 digit passwords for leisure files, which are his most important files.Q. How many leisure passwords are possible? Correct answer is '2'. Can you explain this answer?

    Group QuestionAnswer the following question based on the information given belowDarren Puppet, the richest person in the world, has ensured extra security for his files. He implements a security based on division algorithms in his passwords. The passwords contain n digits such that :I. It contains all digits from 1 to n, and all of them are used exactly once.II. The number formed by the first two digits is divisible by 2, the number formed by the first 3 digits is divisible by 3, and so on such that the n digit password is divisible by n. (For example, 123 is such a 3 digit password, 12 is divisible by 2 and 123 by 3).To remember which password he is using, Darren ranks all the passwords having certain number of digits in increasing order and hence by just knowing the number of digits and the rank, he is able to correctly enter the password. (For example, for 3 digits passwords, 123 has rank 1 and 321 has rank 2)It is known that he uses 6 digit passwords for leisure files, which are his most important files.Q. What is/are the possible values of the fourth digit of the password that he uses for leisure files? Correct answer is '6'. Can you explain this answer?

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    N is a multiple of 72. Also, it is known that all the digits of N are different. What is the largest possible value of N?Correct answer is '9876543120'. Can you explain this answer?
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