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10! is divisible by 3x5y, where x and y are positive integer
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12 Aug 2018, 16:00
12 Aug 2018, 16:00
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Question Stats:
59% (01:16) correct
40% (01:08) wrong based on 103 sessions
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10! is divisible by \(3^x5^y\), where x and y are positive integers.
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Quantity A |
Quantity B |
The greatest possible value for x |
Twice the greatest possible value for y |
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Re: 10! is divisible by 3x5y, where x and y are positive integer
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17 Aug 2018, 16:09
17 Aug 2018, 16:09
4
Expert Reply
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.
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.
Re: 10! is divisible by 3x5y, where x and y are positive integer
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09 Mar 2019, 03:24
09 Mar 2019, 03:24
As the maximum values are x = 4 and y = 2, should not be A greater?
