A natural number is randomly selected from 100 to 360. Find the probability that it is divisible by exactly two of the three numbers 4, 6 and 8.a)11/87b)32/261c)10/87d)34/261Correct answer is option 'A'. Can you explain this answer?

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A natural number is randomly selected from 100 to 360. Find the probability that it is divisible by exactly two of the three numbers 4, 6 and 8.

a)
11/87
b)
32/261
c)
10/87
d)
34/261

Correct answer is option 'A'. Can you explain this answer?

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A natural number is randomly selected from 100 to 360. Find the proba...

Total number of numbers = 360 − 100 + 1 = 261

LCM of 4,6 and 8 is equal to 24

Here, we will take 3 cases.

Case 1: Numbers divisible by 4 and 6 but not 8.

Case 2: Numbers divisible by 6 and 8 but not 4.

Case 3: Numbers divisible by 8 and 4 but not 6.

Case 1: Numbers of the form 12, (12 + 24),(12 + 2 × 24),…

From 100 to 360, the numbers are from (12 + 4 × 24) to (12 + 14 × 24) = (14 − 4 + 1) numbers = 11 numbers.

case 2:

Numbers divisible by 8 have to be divisible by 4

Hence, no number exists in this case.

case 3:

Numbers of the form 8, (8 + 24), (8 + 2 × 24),… and

16, (16 + 24), (16 + 2 × 24),…

From 100 to 360 , the numbers are from (8 + 4 × 24) to (8 + 14 × 24) and (16 + 4 × 24) to (16 + 14 × 24)

= (14 − 4 + 1) + (14 − 4 + 1) = 22

Required probability

Alternative Method:

LCM of 4,6 and 8 is equal to 24

Number of natural numbers that are divisible by 24

Number of natural numbers that are divisible by 8

= (45 − 12 − 11) = 22

Number of natural numbers that are divisible by 4

= (30 − 8 − 11) = 11

Required probability

Hence the correct option is A.

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A natural number is randomly selected from 100 to 360. Find the probability that it is divisible by exactly two of the three numbers 4, 6 and 8.a)11/87b)32/261c)10/87d)34/261Correct answer is option 'A'. Can you explain this answer?

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