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A number is selected at random from the set of natural numbers. The probability that the sum of the digits of its square is 39 is
- a)1/6
- b)0
- c)1/10
- d)1/9
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A number is selected at random from the set of natural numbers. The pr...
Explanation:
Let's assume the number selected is n.
Step 1: Finding the range of n
We know that the sum of the digits of n^2 cannot be greater than the sum of the digits of n*9, i.e., 81. So, the maximum value of n can be found as follows:
n*9 < 10^k="" (where="" k="" is="" the="" number="" of="" digits="" in="" />
9n < />
n < />
Therefore, the maximum value of n is 111, since 10^3/9 = 111.11.
Step 2: Finding the probability
We need to find the probability that the sum of the digits of n^2 is 39. Let's assume that the sum of the digits of n is x. Then, the sum of the digits of n^2 can be written as:
(x^2 - 2s) (where s is the sum of the digits of x^2)
We know that x can range from 1 to 19 (since the maximum value of n is 111). So, we can calculate the value of s for each value of x and find the number of values of s that give a sum of 39.
After calculating, we get that there are no values of s that give a sum of 39. Therefore, the probability is 0.
Hence, the correct answer is option 'B'.
Let's assume the number selected is n.
Step 1: Finding the range of n
We know that the sum of the digits of n^2 cannot be greater than the sum of the digits of n*9, i.e., 81. So, the maximum value of n can be found as follows:
n*9 < 10^k="" (where="" k="" is="" the="" number="" of="" digits="" in="" />
9n < />
n < />
Therefore, the maximum value of n is 111, since 10^3/9 = 111.11.
Step 2: Finding the probability
We need to find the probability that the sum of the digits of n^2 is 39. Let's assume that the sum of the digits of n is x. Then, the sum of the digits of n^2 can be written as:
(x^2 - 2s) (where s is the sum of the digits of x^2)
We know that x can range from 1 to 19 (since the maximum value of n is 111). So, we can calculate the value of s for each value of x and find the number of values of s that give a sum of 39.
After calculating, we get that there are no values of s that give a sum of 39. Therefore, the probability is 0.
Hence, the correct answer is option 'B'.
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