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Number Theory Level 3

If the probability that a number chosen b/w 1 and 1000(both inclusive) is not composite given that it is not an even number be represented by a b \frac{a}{b} where (a,b)=1 ,find the sum of the digits of 4 ∣ a − b ∣ 4|a-b| ?

9 10 7 8

Vinay Sipani by Vinay Sipani , 26, India

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2 solutions

Vinay Sipani
May 25, 2014

There are 500 odd numbers = > d e n o m i n a t o r = 500 =>denominator=500

There are 168 prime numbers less than 1000 but 2 is not considered as it is not an odd number. Hence,167 prime numbers are considered. Here, 1 is also considered as it is neither prime nor composite. = > N u m e r a t o r = 168 =>Numerator=168

Also,it is given that a and b are co-prime => a b = 42 / 125 \frac{a}{b}=42/125 = > 4 ∣ a − b ∣ = 332 =>4|a-b|=332

Therefore,sum of the digits=8

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Sudhir Aripirala
Jan 26, 2015

Silly solution. Given answer is in the form of 4(a-b). Therefore, answer must be a multiple of 4 in given options. Answer=8

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