GATE | GATE CS 2010 | Question 27

What is the probability that divisor of 10⁹⁹ is a multiple of 10⁹⁶?
(A) 1/625
(B) 4/625
(C) 12/625
(D) 16/625

Answer: (A)
Explanation: Following are multiples of 10⁹⁶ which are also divisor of 10⁹⁹

10⁹⁶, 2×10⁹⁶, 4×10⁹⁶, 5×10⁹⁶, 8×10⁹⁶, 10×10⁹⁶, 20×10⁹⁶, 25×10⁹⁶, 40×10⁹⁶, 50×10⁹⁶, 100×10⁹⁶, 125×10⁹⁶, 200×10⁹⁶, 250×10⁹⁶, 500×10⁹⁶, 1000×10⁹⁶

The total number of divisors of 10⁹⁹ = 10000  (See http://www.math.cmu.edu/~mlavrov/arml/13-14/number-theory-09-29-13.pdf)

So the probability = 16/10000 = 1/625
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