Question

What is the probability of choosing at random number between 1 and 64 which is 7-Smooth? Explain

Answer 1

7-Smooth numbers are those numbers whose maximum prime factor is less than or equals to 7. For example 42 = 2 x 3 x 7 is 7-Smooth but 52 = 2 x 2 x 13 is not. Note that 12 = 2 x 2 x 3 is also 7-Smooth as 3 is lesser than 7.

To find non-7-Smooth numbers, we need to find out the prime numbers greater than 7 and those numbers which can be divisible by primes greater than 7.

Prime numbers greater than 7 between 1 and 64 are,

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, total 14.

Numbers between 1 and 64 and divisible by

11 - 11, 22, 33, 44, 55, total 4 excluding 11.

13 - 13, 26, 39, 52, total 3 excluding 13.

17 - 17, 34, 51, total 2 excluding 17.

19 - 19, 38, 57, total 2 excluding 19.

23 - 23, 46, total 1 excluding 23.

29 - 29, 58, total 1 excluding 29.

31 - 31, 62, total 1 excluding 31.

So, total non-7-Smooth numbers are 14 + 4 + 3 + 2 + 2 + 1 + 1 + 1 = 28.

We take 1 as non-7-Smooth so total becomes 29. So, the probability of choosing a non-7-Smooth number is 29/64.

So, the probability of choosing a 7-Smooth number is 1 - 29/64 = 35/64.