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Q.

The largest non - negative number k such that 24k divides 12! is 


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a

2

b

3

c

4

d

5 

answer is B.

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Detailed Solution

We know that if there are two numbers p and q which are relative prime (it means greatest common factor (GCD) between p and q is 1) and they exactly divide any number n if and only if their product pq also exactly divides the number n. It is one of the divisibility rules. We know that the largest power on any prime p is r such that pr divides n! then we have,
r = np + np2 + np3 + ...
Here [x] is the greatest integer function which returns greatest integer less than equal to x. Now, let us consider the prime factorization of 24. We have
24 = 2×2×2×3 = 23×3 = 8 × 3
If a number is divisible by the relative primes 8 and 3 then the number is also divisible by 24. Let us consider some natural numbers λ
24k = 2m3lλ......(1)
Here m and l are the largest power on 2 and 3 respectively. If 24k divide 12! then 2m and 3l also divide 12! . We use largest power formula for n=12 and p=2 to get m as,
m = 122 + 1222 + 1223 + 1224  ...
⇒ m = [6] + [3] + [1.5] + [0.75] + ...
⇒ m = 6 + 3 + 1 + 0 + 0...
⇒ m = 10
We use largest power formula for n=12 and p=3 to get ll as,
l = 123 + 1232 + 1233 + ...
⇒ l = [4] + [1.3̲] + [0.4̲] +...
⇒ l = 4 + 1 + 0...
⇒ l = 5 
We put the value of m,l in equation (1) to have,
24k = 21035λ
⇒ 24k = (2×2×2×2×2×2×2×2×2×2×2×3×3×3×3×3)λ
We consider the divisibility rule of 24 and arrange the product 24 = 8×3 = 23 × 3 will be close to each other. We have,
⇒ 24k = 23× 3 × 23× 3 × 23× 3 × 23× 3 × λ
⇒ 24k = (23× 3 )3 × 18λ
⇒ 24k = 243 × 18λ        We compare power on 24 on both sides of the equation and find the largest power on k such that 24k divides 12! as 3.
So, the correct answer is “Option 2”.
 
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