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Question 11 : The probability that a randomly chosen factor of 10¹⁹ is a multiple of 10¹⁵ is

1. \\frac{1}{25})
2. \\frac{1}{12})
3. \\frac{1}{20})
4. \\frac{1}{16})

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Video Explanation

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Explanatory Answer

10¹⁹ = 2¹⁹ × 5¹⁹
This is of the form 2^x × 5^y
x can take values from 0 to 19
y can take values from 0 to 19
x can take 20 vales and y can also take 20 values
Chosen number has 20 × 20 = 400 factors

Among that we need a number which is a multiple of 10¹⁵ = 2¹⁵ × 5¹⁵
This is of the form 2^a × 5^b

We need a number that is not only a multiple of 2^a × 5^bbut also a factor of 2¹⁹ × 5¹⁹

Looking for numbers which are in form 2^p × 5^q

Since 2^p × 5^q is a factor of 2¹⁹ × 5¹⁹, p and q should be less than or equal to 19
Since 2^p × 5^q is a multiple of 2¹⁵ × 5¹⁵, p and q should be more than or equal to 15

p and q can take any of the 5 values 15, 16, 17, 18, 19.
So, the probability is \\frac{5 × 5}{20 × 20}) = \\frac{1}{16})

The question is " The probability that a randomly chosen factor of 10¹⁹ is a multiple of 10¹⁵ is "

Hence, the answer is \\frac{1}{16})

Choice C is the correct answer.

 

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