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What is the smallest positive integer that is non-prime

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What is the smallest positive integer that is non-prime [#permalink] New post 09 Sep 2018, 20:59
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What is the smallest positive integer that is non-prime and not a factor of \(9!\) ?

[Reveal] Spoiler: OA
22

Last edited by Carcass on 10 Sep 2018, 12:18, edited 1 time in total.
Edited by Carcass
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 10 Sep 2018, 05:11
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AchyuthReddy wrote:
What is the smallest positive integer that is non-prime and not a factor of 9!?


-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------


9! = (9)(8)(7)(6)(5)(4)(3)(2)(1)
So, 1 to 9 are definitely factors of 9!
10 is also a factor of 9! since 9! = (9)(8)(7)(6)(5)(4)(3)(2)(1) = (9)(8)(7)(6)(10)(4)(3)(1)
11 is prime, so we can ignore that.
12 is also a factor of 9! since 9! = (9)(8)(7)(6)(5)(4)(3)(2)(1) = (9)(8)(7)(6)(12)(4)(3)(1)

Using the same logic, we can show that 14, 15, 16, 18, 20 and 21 are all factors of 9!

However, 22 is NOT a factor of 9!
We know this because 22 = (2)(11) and there is no 11 hiding in the prime factorization of 9!

Answer: 22

Cheers,
Brent
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 14 Oct 2019, 10:26
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Hi Brent,

Is there any other way to solve this? I could not understand the process.

Thanks.
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 25 Dec 2019, 08:32
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Salina wrote:
Hi Brent,

Is there any other way to solve this? I could not understand the process.

Thanks.


How about this?

9! = 9 x 8 x 7 x 6 x 5 x 4 x 2 x 1
= (3 x 3) x (2 x 2 x 2) x 7 x (2 x 3) x 5 x (2 x 2) x 2 x 1
= (2⁷)(3³)(5)(7)

Once we know the prime factorization we can see that:
1, 2, 3, 4, 5, 6, 7, 8, and 9 are all factors of 9!
Also, 10 is a factor of 9! since 10 = (5)(2), and we can see one 5 and one 2 hiding in the prime factorization of 9!
Also, 12 is a factor of 9! since 12 = (2)(2)(3), and we can see two 2's and one 3 hiding in the prime factorization of 9!
14 is a factor of 9! since 12 = (2)(7), and we can see one 2's and one 7 hiding in the prime factorization of 9!
16 is a factor of 9! since 16 = (2)(2)(2)(2), and we can see four 2's hiding in the prime factorization of 9!
18 is a factor of 9! since 18 = (2)(3)(3), and we can see one 2 and two 3's hiding in the prime factorization of 9!
20 is a factor of 9! since 12 = (2)(2)(5), and we can see two 2's and one 5 hiding in the pprime factorization of 9!
22 is NOT a factor of 9! since 22 = (2)(11), and we there are NO 11's hiding in the prime factorization of 9!

Does that help?

Cheers,
Brent
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 10 Apr 2020, 06:03
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Hi,
i think there is another option, maybe faster for some people

we know that 11 is the first prime number that isn't in 9!
and we have the rule : if K isn't a divisor of N, then JK isn't a divisor on N.
implementing this rule, we just need to take 11 and multiple it with the first number bigger then 1 (to make the number not prime) => 2X11 = 22.
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 29 May 2020, 15:30
AchyuthReddy wrote:
What is the smallest positive integer that is non-prime and not a factor of \(9!\) ?

[Reveal] Spoiler: OA
22


I'm not sure, but i think if we find prime number after the n, and multyply by 2 which comes smallest positive, not prime and not factor of n!.

here ,prime number after 9 is 11, so multiply 11 with 2==22.

another example:
if question asks for 50! ,
next prime number after 50 is 53. and multiply by 53*2==106, which is the smallest positive, not prime and not factor of 50!.

i'll appreciate,if someone correct me, if i'm wrong.
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Re: What is the smallest positive integer that is non-prime [#permalink] New post 01 Jun 2020, 05:58
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Re: What is the smallest positive integer that is non-prime   [#permalink] 01 Jun 2020, 05:58
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