9600 could be written as 12 * 800
Now 10 is a factor of 800, so is 20, so is 400, including many others.
10, 20 and 400 are not divisible by 12 but (12*10), (12*20) and (12*400) are divisible by 12 because we are multiplying and dividing my 12. Hence every factor of 800 is also a factor of 9600 and is divisible by 12 if we are multiplying each factor of 800 by 12.
To find the number of factors of 800 is a straightforward application of number of factors formula:-
(p+1)(q+1)(r+1)... [where p,q,r are exponents of each prime factor]
Therefore 800 can be written as \(2^5∗5^2\)
Therefore the number of factors of 800 are (5+1)(2+1) = 6*3 = 18 factors.
Therefore the no. of factors of 9600 which are divisible by 12 are 18 factors in total.
But the question stem asks us how many factors of 9600 are NOT divisible by 12?
Therefore we will have to deduct all the factors of 9600 which are divisible by 12 from all the factors of 9600 to get all the factors of 9600 which are NOT divisible by 12.
9600 can be written as \(2^7∗3*5^2\)
Therefore the number of factors of 9600 are (7+1)(1+1)(2+1) = 8*2*3 = 48 factors.
All the factors of 9600 which are NOT divisible by 12 = 48 factors - 18 factors = 30 factors [hence the correct answer is (B)]
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