1. 5
the worst case: asking the whether the number can be divided by 2,3,5,7 or 11.
2. 53/23
if the minimum divisor is 2, need to ask only 1 question, we have [162/2]=81 such numbers;
if the minimum divisor is 3, minimum 2 questions, we have [162/2]-[162/6]=27 such numbers;
similarly, for minimum divisor of 5, 11 numbers and 3 questions; for minimum divisor of 7, 7 numbers and 4 questions; for minimum divisor of 11, 3 numbers and 5 questions; for prime numbers greater than 11, 32 numbers and 5 questions.
so the expected value is (81*2+27*2+11*3+7*4+3*5+32*5)/161 = 371/161 = 53/23
the worst case: asking the whether the number can be divided by 2,3,5,7 or 11.
2. 53/23
if the minimum divisor is 2, need to ask only 1 question, we have [162/2]=81 such numbers;
if the minimum divisor is 3, minimum 2 questions, we have [162/2]-[162/6]=27 such numbers;
similarly, for minimum divisor of 5, 11 numbers and 3 questions; for minimum divisor of 7, 7 numbers and 4 questions; for minimum divisor of 11, 3 numbers and 5 questions; for prime numbers greater than 11, 32 numbers and 5 questions.
so the expected value is (81*2+27*2+11*3+7*4+3*5+32*5)/161 = 371/161 = 53/23
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