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For any integer n greater than 1, n* denotes the product of

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galiya
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For any integer n greater than 1, n* denotes the product of [#permalink] New post   23 Apr 2012, 11:51
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For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25

Could please someone explain me the logic of the solution for this task since the official explanation isn't clear for me
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E
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Bunuel
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   23 Apr 2012, 12:31
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Galiya wrote:
For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25

Could please someone explain me the logic of the solution for this task since the official explanation isn't clear for me


"n* denotes the product of all the integers from 1 to n, inclusive" means that n*=1*2*3*...*n=n!.

Which means that, 4*=4!=24 and 5*=5!=120. So, the question basically asks "how many multiples of 4 are there between 24 and 120, inclusive"

# of multiples of 4 between 24 and 120, inclusive is (last-first)/multiple+1=(120-24)/4+1=25 (check this: how-many-multiples-of-4-are-there-between-12-and-94862.html).

Answer: E.

Hope it helps.
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galiya
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   23 Apr 2012, 12:42
I'm familiar w this formula, but i need to grasp it
subtracting 24 from 120 we eliminate some multiples of four between 24 and 120: 100, 104, ...112...
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   23 Apr 2012, 13:16
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Galiya wrote:
I'm familiar w this formula, but i need to grasp it
subtracting 24 from 120 we eliminate some multiples of four between 24 and 120: 100, 104, ...112...


You can derive it from arithmetic progression formula.

Multiples of 4 represent evenly spaced set (aka arithmetic progression).

Now, if we have \(n\) terms in arithmetic progression, the first term is \(a_1\) and the common difference of successive members is \(d\), then the \(n_{th}\) term of the sequence is given by: \(a_ n=a_1+d(n-1)\) --> \(n=\frac{a_ n-a_1}{d}+1\).

You can see that it's basically the same formula as in my previous post.
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   23 Apr 2012, 16:26
The more I browse the more tricks I learn, this question would have taken me a good 3 minutes. Thanks for the good explanation.
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   11 Nov 2013, 21:09
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Galiya wrote:
For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25

Could please someone explain me the logic of the solution for this task since the official explanation isn't clear for me


here, 4* = 4! and 5* = 5!

question is asking no. of terms in an AP whose first term is 24(4!), last term is 120(5!) and common difference is 4.

last term OR nth term = first term +(no. of tems -1)(common difference)
Tn = a + (n-1)d

120 = 24 + (n-1)4
96/4 = n-1

n = 25 .....
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   13 Mar 2017, 00:26
4*=4!=24
5*=5!=120

multiples of b/w 24 and 120= (120-24)/4+1=24+1=25
answer E
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   15 Mar 2017, 16:49
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Galiya wrote:
For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25


We are given that for any integer n greater than 1, n* denotes the product of all the integers from 1 to n inclusive. Thus the notation n* is same as the notation n!.

Thus, 4* = 4! = 24 and 5* = 5! = 120.

We need to determine how many multiples of 4 are between 24 and 120.

(largest multiple of 4 in the set - smallest multiple of 4 in the set)/4 + 1

(120 - 24)/4 + 1 = 96/4 + 1 = 25

Answer: E
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   13 Jan 2022, 02:22
galiya wrote:
For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25

Could please someone explain me the logic of the solution for this task since the official explanation isn't clear for me


An alternative approach, which is practically an application of the same principles as @Bunuel's solution but just reduces little arithmetic.

\frac{(5!-4!)}{4} + 1
\frac{2*3*4(5-1)}{4} + 1
= 25
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For any integer n greater than 1, n* denotes the product of [#permalink] New post   13 Jan 2022, 02:23
galiya wrote:
For any integer n greater than 1, n* denotes the product of all the integers from 1 to n, inclusive. How many multiples of 4 are there between 4* and 5*, inclusive?

A. 5
B. 6
C. 20
D. 24
E. 25

Could please someone explain me the logic of the solution for this task since the official explanation isn't clear for me


An alternative approach, which is practically an application of the same principles as @Bunuel's solution but just reduces little arithmetic and will be suitable for larger numbers.

(5!-4!)/4 + 1
=[2*3*4(5-1)]/4 + 1
= 25
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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   02 Sep 2022, 22:02
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Given that n∗ denotes the product of all the integers from 1 to n, inclusive and we need to find How many multiples of 4 are there between 4* and 5*, inclusive

4* = Product of all the integers from 1 to 4, inclusive = 1*2*3*4 = 4*1*2*3 = 4*6
5* = Product of all the integers from 1 to 5, inclusive = 1*2*3*4*5 = 4*1*2*3*5 = 4*30

So, number of multiples of 4 from 4* to 5* inclusive = Number of multiples of 4 from 4*6 to 4*30 inclusive = (30-6)+1 = 24 + 1 = 25

So, Answer will be E.
Hope it helps!

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Re: For any integer n greater than 1, n* denotes the product of [#permalink] New post   30 May 2023, 05:00
In this question, the symbol * is interchangeable with the
concept of "factorial", usually abbreviated with "!". Thus, 4* = 4x3x2x1 =
24, and 5* = 5 x 4 x 3 x 2 x 1 = 120. The question boils down to: how many
multiples of 4 are there between 24 and 120, inclusive. For that, find the difference: 120-24 = 96. Divide by 4: 96/4 = 24.
Finally, add 1, because the endpoints are both multiples of 4, and the question uses the
word "inclusive": 24 + 1 = 25, choice (E).
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Re: For any integer n greater than 1, n* denotes the product of [#permalink]
30 May 2023, 05:00
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