Explanation

First, expand 10! as 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

(Do not multiply all of those numbers together to get 3,628,800—it’s true that 3,628,800 is the value of 10!, but analysis of the prime factors of 10! is easier in the current form.)

Note that 10! is divisible by 3x5y, and the question asks for the greatest possible values of x and y, which is equivalent to asking, “What is the maximum number of times you can divide 3 and 5, respectively, out of 10! while still getting an integer answer?”

In the product 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, only the multiples of 3 have 3 in their prime factors, and only the multiples of 5 have 5 in their prime factors. Here are all the primes contained in 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 and therefore in 10!:

10 = 5 × 2
9 = 3 × 3
8 = 2 × 2 × 2
7 = 7
6 = 2 × 3
5 = 5
4 = 2 × 2
3 = 3
2 = 2
1 = no primes

There are four 3’s and two 5’s total. The maximum values are x = 4 and y = 2. Therefore, the two quantities are equal.
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As the maximum values are x = 4 and y = 2, should not be A greater?

\(10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1\)

\((2*5) * (3^2) * (2^3) * 7 * (2*3) * 5 * 3 * (2^2) * 1\)

\(\frac{7 * 5^2 * 3^4 * 2^7 * 1}{3^x 5^y}\)

As you clearly see the quantity must be divided by \(3^x\) and \(5^y\).

In the numerator \(3^4\) and \(5^2\) , which means that the exponent of 3 is \(4 = x\) and the exponent of 5 is \(2=y\)

A > B

I think also the answer should be A



Hope this helps.

Regards

Be careful.
Quantity B = TWICE the greatest possible value for y

Cheers,
Brent
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Thanks to both of you, it is clear now.

I often make this kind of silly mistakes, I need to read more carefully.

Thanks again!

Trueeeeeee

x = 4 and y= 2 but B is twice. So, y = 4

C is the answer.

Thank you Sir

Simply write down the factorial value of 10! i.e.
10.9.8.7.6.5.4.3.2.1
Then write down prime factors of above and add power values of 2 and 3, from which x=4 and y=2.
Therefore A and B are equal.
Answer is C

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If the question was 100! we cannot do from traditional method (100 X 99 X 98... X 1)
Better solution:
100/3 -> 33
100/9 -> 11
100/27 -> 3
100/81 -> 1
Add all -> 33+11+3+1 = 48 (x=48)

100/5 -> 20
100/25 -> 4
Add all -> 24 (y=24)

X=2Y

Similarly for 10!
10/3 -> 3
10/9 -> 1
Add all -> 4 (x = 4)

10/5 -> 2 (y = 2)

X=2Y