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Bunuel
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For any integer P greater than 1, P! denotes the product of all the in 18 Apr 2023, 01:30
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65% (hard)

Question Stats:

62% (02:41) correct 38% (02:46) wrong based on 233 sessions

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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 10080*h and h is a perfect square, what is the greatest possible value of h?

A. 72
B. 36
C. 9
D. 8
E. 4


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B
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For any integer P greater than 1, P! denotes the product of all the in 18 Apr 2023, 02:31
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If 10! Is divisible by 10080*h and h is a perfect square, what is the greatest possible value of h?

10080 = 2 * 7 * 8 * 9 * 10
=> h has to be a product of either of these numbers - (3, 4, 5, 6)
the greatest possible value of h is 36 as 36 is a perfect square.
Answer B
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For any integer P greater than 1, P! denotes the product of all the in 18 Apr 2023, 04:31
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KarishmaB could you please explain this one?

Thank you in advance.
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KarishmaB
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For any integer P greater than 1, P! denotes the product of all the in 19 Apr 2023, 03:12
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Bunuel
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 10080*h and h is a perfect square, what is the greatest possible value of h?

A. 72
B. 36
C. 9
D. 8
E. 4


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\(10080 = 2^5 * 3^2 *5 * 7\)

"10! Is divisible by 10080*h" means

\(\frac{10!}{10080h}=x\) where x is an integer.

\(10! = 10080 * h * x\)

\(1*2*3*4*5*6*7*8*9*10 = 2^5 * 3^2 *5 * 7 * h * x\)

Look at all that gets cancelled:

1*2*3*4*5*6*7*8*9*10 = 2^5 * 3^2 *5 * 7 * h * x

We are left with:

\(2 * 3 * 6 * 10 = h * x = 2^2 * 3^2 * 2 * 5\)

Since h is a perfect square, its greatest value is 2^2 * 3^2 since a perfect square must have even exponents for all prime numbers.

Hence the greatest value of h is 36.

Answer (B)
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For any integer P greater than 1, P! denotes the product of all the in 22 Nov 2024, 22:59
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Question is asking 10!/1008*H where H is a square

Basically, 1008*H can have every factor of 10! for 10! to be able to still be divisible by 1008*H

1008 = 2^5 * 3^2 * 5^1 * 7^1
10! = 2^7 * 3^4 * 5^1 * 7^1

Extra numbers in 10! = 2^2*3^2 = 6^2 = 36 which is a perfect square
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