Answer:
975634
Explanation:
You want the largest possible multiple of 11 that can be made from the digits 3,4,5,6,7,9.
Divisibility by 11
A number will be divisible by 11 if the difference between the sum of its even digits and the sum of its odd digits is a multiple of 11. For a 6-digit number, the sum of 3 of the digits must differ from the sum of the other 3 by 0 or 11.
Sums
The sum of all of the given digits is 34, an even number. If we can divide the set of digits into groups of 3 that each total 17, then we can form the required number. (There is no way to form groups whose sums differ by 11.)
The number 9 must be in one of the groups, so the other two numbers in that group must total 8. The only such pair is 5 and 3.
The first digit will be 9, and alternate digits after that will be 5 and 3. The remaining digits are 7, 6, and 4. Putting these in decreasing order will give the largest number:
975634 = 11×88694
The largest number divisible by 11 is 975634.
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