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the highest prime less than 50, that divides the binomial coefficient 100C50 ? a) 37, b) 31, c)47 d)43

the highest prime less than 50, that divides the binomial coefficient 100C50 ? a) 37, b) 31, c)47 d)43

Grade:12

1 Answers

Jit Mitra
25 Points
9 years ago

100C50 = 100!/(50!)2

 

We can express the factorials as exponents of prime numbers. But that is not suppossedly required. We start with the highest ones.

 

Exponent of 47 in 100! = [100/47] = 2

Exponent of 47 in (50!)^2  =  [50/47]*2 = 1*2 = 2

So the exponent of 47 in (50!)^2 and 100! is same, hence they cancel out.

 

(For mcq purposes you can make an intelligent guess from right his point. You require a number that divides 100 thrice but 50 once. 33 is the required number).

Exponent of 43 in 100! = [100/43] = 2

Exponent of 43 in (50!)^2  =  [50/43]*2 = 1*2 = 2

 

Exponent of 37 in 100! = [100/37] = 2

Exponent of 37 in (50!)^2  =  [50/37]*2 = 1*2 = 2

 

Exponent of 33 in 100! = [100/33] = 3

Exponent of 33 in (50!)^2  =  [50/33]*2 = 1*2 = 2

 

Here, one 33 remains uncancelled in the numerator. So the answer is actually 33. But as i dont see an option, i would go for 31 because the other numbers dont satisfy the condition whereas 31 does. But it isnt the HIGHEST prime number doing that.

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