Question

In a nuclear power plant, a technician is allowed for an interval of maximum 100 minutes. A timer with a bell rings at specific intervals of time such that the minutes when the timer rings are not divisible by 2, 3, 5 and 7. The last alarm rings with a buzzer to give time for decontamination of the technician. How many times will the bell ring within these 100 minutes and what is the value of the last minute when the bell rings for the last time in a 100 minute shift?

• A. 25 times, 89
• B. 21 times, 97
• C. 22 times, 97
• D. 19 times, 97

Answer 1

The correct option is C 22 times, 97

In order to find how many times the alarm rings we need to find the number of numbers below 100 which are not divisible by 2, 3, 5 or 7. This can be found by 100 - (numbers divisible by 2) - (numbers divisible by 3 but not by 2) - (numbers divisible by 7 but not by 2 or 3 or 5).
Numbers divisible by 2 up to 100 would be represented by the series 2, 4, 6, 8, 10 .. 100 $\to$ A total of 50 numbers.
Numbers divisible by 3 but not by 2 up to 100 would be reprsented by the series 3, 9, 15, 21 ... 99 finding the number of number in this series:
[(last term - first term)/ common difference] + 1 = [99 - 3)/6] + 1 = 16 + 1 =17.
Numbers divisible by 5 but not by 2 or 3: Numbers divisible by 5 but not by 2 up to 100 would be represented by the series 5, 15, 25, 35 ... 95 $\to$ A total of 10 numbers. But from these numbers, the numbers 15, 45 and 75 are also divisible by 3. Thus,
we are left with 10 - 3 = 7 new numbers which are divisible by 5 but not by 2 and 3.
Numbers divisible by 7, but not by 2, 3 or 5:
Numbers divisible by 7 but not by 2 up to 100 would be represented by the series 7, 21, 35, 49, 63, 77, 91 $\to$ A total of 7 numbers. But from these numbers, we should not count 21, 35 and 63 as they are divisible by either 3 or 5. Thus a total of 7 - 3 = 4 numbers are divisible by 7 but not by 2, 3 or 5.