The largest two digit prime factor of $200 C 100$ is

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None of these

answer is A.

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Detailed Solution

Prime number is a number which is divisible just by itself and one.
A prime factor of a number is a prime number which divides the given number.
Here, we are asked to find the largest prime factor of

200 C 100

If a number is given in the form

n C r

then it can be written as

n C r = n! r! (n − r)!

Comparing this with our given number we get,

200 C 100 = 200! 100! (200 − 100)! ⇒ 200 C 100 = 200! 100! 100!

If p is a two-digit prime factor of 100, between 1 and 100, then it will appear twice in the denominator.
So for p to be the prime factor

200 C 100

, it should appear thrice in the numerator such that,
3p < 200
Now we compare the Options 61, 59, 17 and check which number gives the largest value when multiplied by three.
Multiplying 3 with the Options we get,
3×61 = 18
3×59 =177
3×17 = 51
We observe that 3×61 gives the highest value, so the largest value of p is 61.

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