Question

The largest two digit prime factor of \[^{200}{C_{100}}\] is
A. \[61\]
B. \[59\]
C. \[17\]
D.None of these

Answer 1

Hint: Recall what is meant by a prime number and what is a prime factor. Break the given number into fraction. Check for the possibilities of a prime factor for the given number. Compare it with the given options and select the correct option.

Complete step-by-step answer:
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
 \[^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Comparing this with our given number we get,
 \[^{200}{C_{100}} = \dfrac{{200!}}{{100!\left( {200 - 100} \right)!}} \\
  { \Rightarrow ^{200}}{C_{100}} = \dfrac{{200!}}{{100!100!}} \]
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,
 \[3 \times 61 = 183 \\
  3 \times 59 = 177 \\
  3 \times 17 = 51 \]
We observe that \[3 \times 61\] gives the highest value, so the largest value of \[p\] is \[61\] .
Hence, the correct answer is option (A) \[61\] .
So, the correct answer is “Option A”.

Note: Numbers can be classified into two types on the basis of their factors; prime number and composite number. A prime number is a number which is divisible just by one and itself, that is they have only two factors, whereas a composite number has more than two factors. Even numbers are considered as composite but two is the only even number which is prime number.