A divisor of N= 253453 is selected at random. The probability that the selected number is divisible by 200 if it is known that it is divisible by 100 is
A) 3/4 B) 1/2 C) 1/4 D) 4/5
A divisor of N= 253453 is selected at random. The probability that the selected number is divisible by 200 if it is known that it is divisible by 100 is
A) 3/4 B) 1/2 C) 1/4 D) 4/5
Firstly, we have to find the number of ways to select a divisor of N. N = 2^5 ´ 3^4´ 5^3
Any divisor of n will be of the form 2^a´ 3^b ´ 5^c
where 0 ≤ a ≤ 5, 0 ≤ b ≤ 4, 0 ≤ c ≤ 3 . For any distinct choices of a,b and c , we get a divisor of N
Total number of divisors = (5 + 1) (4 + 1) (3 + 1) = 120.
Here (5+1) means that there are 5+1, that is 6 ways to select a number of the form 2^5 as 'a' can be 0,1,2,3,4 or 5.
Similarily for 3^4 and 5^3.
Now,
It is mentioned in the question that the selected divisor is divisible by 100.
As the divisor is of the form 2^a 3^b 5^c, then for the divisor to be divisible by 100 i.e. 2^2 5^2 the divisor must contain atleast 2^2×5^2.
So, from all the 120 selected divisors above, the ones having 2^2×5^2 can be determined as follows,
Given the constraints,
'a' can be between 2 and 5 i.e. 2<=a<=5 which means 4 ways, and so
0<= b <=4 which means 5 ways and 2<= c <=3 which means 2 ways
So, number of divisors divisble by 100 = 4×5×2 = 40
Now, we have selected the divisors which are divisble by 100 which is the total number of outcomes of the probability.
The total number of outcomes = 40
Now, the divisors which are disivible by 200 will also be divisble by 100.And out of the 40 some will still be left who are not divisible by 200.So, if we find out the divisors which are divisible by 200 we can calculate the probability.
No. of divisors divisible by 200 i.e.2^3×5^2 = (3×5×2) = 30 which is the possible outcomes.
So, the probability = possible outcomes/total outcomes
P = 30/40 = 3/4
Hence, (a) is the correct answer.
Hit like if you liked the solution.
N = 2^5 ´ 3^4´ 5^3
Any divisor of n will be of the form 2^a´ 3^b ´ 5^c
where 0 ≤ a ≤ 5, 0 ≤ b ≤ 4, 0 ≤ c ≤ 3 . For any distinct choices of a,b and c , we get a divisor of N
Total number of divisors = (5 + 1) (4 + 1) (3 + 1) = 120.
Here (5+1) means that there are 5+1, that is 6 ways to select a number of the form 2^5 as 'a' can be 0,1,2,3,4 or 5.
Similarily for 3^4 and 5^3.
Now,
It is mentioned in the question that the selected divisor is divisible by 100.
As the divisor is of the form 2^a 3^b 5^c, then for the divisor to be divisible by 100 i.e. 2^2 5^2 the divisor must contain atleast 2^2×5^2.
So, from all the 120 selected divisors above, the ones having 2^2×5^2 can be determined as follows,
Given the constraints,
'a' can be between 2 and 5 i.e. 2<=a<=5 which means 4 ways, and so
0<= b <=4 which means 5 ways and 2<= c <=3 which means 2 ways
So, number of divisors divisble by 100 = 4×5×2 = 40
Now, we have selected the divisors which are divisble by 100 which is the total number of outcomes of the probability.
The total number of outcomes = 40
Now, the divisors which are disivible by 200 will also be divisble by 100.And out of the 40 some will still be left who are not divisible by 200.So, if we find out the divisors which are divisible by 200 we can calculate the probability.
No. of divisors divisible by 200 i.e.2^3×5^2 = (3×5×2) = 30 which is the possible outcomes.
So, the probability = possible outcomes/total outcomes
P = 30/40 = 3/4
Hence, (a) is the correct answer.
Hit like if you liked the solution.
