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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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Explanatory Answer

1019 = 219 × 519
This is of the form 2x × 5y
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 2a × 5b

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

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

Since 2p × 5q is a factor of 219 × 519, p and q should be less than or equal to 19
Since 2p × 5q 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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