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Let S be the set of all 6 -digit numbers formed using only the digits a and b , where a, b 1, 2, , 9 and a < b . It is given that every number in S is divisible by 7 . If the probability that a randomly chosen number from S is a multiple of 9 is 5 16 , then the value of a^2 + b^2 is equal to

Correct answer

65

Step-by-step solution

The set S contains 2^6 = 64 numbers. Since every number in S is divisible by 7 , the difference between any two numbers in S must also be divisible by 7 . Consider the numbers aaaaab and aaaaba . Their difference is 10(a - b) - (a - b) = 9(a - b) . For 9(a - b) to be divisible by 7 , since 9 and 7 are coprime, a - b must be a multiple of 7 . Given a, b 1, 2, , 9 and a Let us check the pair (1, 8) : A number in S has k eights and (6 - k) ones, where 0 k 6 . The sum of the digits is 8k + 1(6 - k) = 7k + 6 . For the n

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