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For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. If 10! Is divisible by 10080h and h
is a perfect square, what is the greatest possible value of h?
is a perfect square, what is the greatest possible value of h?
- a)72
- b)36
- c)9
- d)8
- e)4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
For any integer P greater than 1, P! denotes the product of all the in...
Given:
- For a positive integer P, P! = P(P-1)(P-2) . . . * 3* 2 * 1
- 10! Is divisible by 10080*h
- 10! = 10*9*8*7*6*5*4*3*2*1
- Since 10! Is divisible by h, h is definitely non-zero (since division by 0 is not defined)
- h is a perfect square
- All perfect squares are non-negative
- Since we’ve inferred above that h is non-zero, h must be a positive integer
To Find: The greatest possible value of h
Approach:
- We are given that 10! Is divisible by 10080*h
2 So, we will first find the value of 
3. Then, we will prime- factorize this value to determine the greatest perfect square that divides this value.
Working out:
- Find the greatest possible value of h
- 6*5*4*3 is divisible by perfect square h
- Writing the prime-factorized form of 6*5*4*3
- 6*5*4*3 =(3*2)*5*22*3 =23*32*5
- The greatest perfect square that divides 23*32*5 is 22 * 32
- So, the greatest possible value of h =22 * 32 = 36
Looking at the answer choices, we see that the correct answer is Option B
Most Upvoted Answer
For any integer P greater than 1, P! denotes the product of all the in...
Explanation:
Given:
- P! denotes the product of all the integers from 1 to P
- 10! is divisible by 10080h
- h is a perfect square
To find:
Greatest possible value of h
Approach:
- 10080 = 2^4 * 3^2 * 5 * 7
- To find the greatest possible value of h, we need to find the highest power of 2, 3, 5, and 7 that can divide 10!
Calculations:
- The highest power of 2 that divides 10! is 2^8 (as there are 8 multiples of 2 in 10!)
- The highest power of 3 that divides 10! is 3^4 (as there are 4 multiples of 3 in 10!)
- The highest power of 5 that divides 10! is 5^2 (as there are 2 multiples of 5 in 10!)
- The highest power of 7 that divides 10! is 7^1 (as there is 1 multiple of 7 in 10!)
- Therefore, the highest power of h = 2^8 * 3^4 * 5^2 * 7^1 = 36^2
Conclusion:
The greatest possible value of h is 36, which is a perfect square, hence the correct answer is option B - 36.
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