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# S97-10

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Math Expert
Joined: 02 Sep 2009
Posts: 51158

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16 Sep 2014, 00:51
1
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Difficulty:

95% (hard)

Question Stats:

42% (01:31) correct 58% (01:26) wrong based on 106 sessions

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If $$n$$ is a positive integer greater than 1, then $$p(n)$$ represents the product of all the prime numbers less than or equal to $$n$$. The second smallest prime factor of $$p(12) + 11$$ is

A. 2
B. 11
C. 13
D. 17
E. 211

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Joined: 02 Sep 2009
Posts: 51158

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16 Sep 2014, 00:51
Official Solution:

If $$n$$ is a positive integer greater than 1, then $$p(n)$$ represents the product of all the prime numbers less than or equal to $$n$$. The second smallest prime factor of $$p(12) + 11$$ is

A. 2
B. 11
C. 13
D. 17
E. 211

The quantity $$p(12)$$ equals the product of all the primes less than or equal to 12. Thus, the number we are looking for is this:
$$2 \times 3 \times 5 \times 7 \times 11 + 11$$

$$= 11 \times (2 \times 3 \times 5 \times 7+1)$$ [factor out the 11]

$$= 11 \times (210 + 1)$$ [do the arithmetic]
$$= 11 \times 211$$
$$= \text{some large number}$$

We want to keep this number factored, and in fact we need to find its prime factorization. So, we ask: is 211 prime? Well, 211 cannot be divided evenly by 2, 3, 5, or 7, because 211 equals a multiple of all those numbers (210), plus 1. The "plus 1" means that 211 won't be divisible by any of the same factors as 210 (except for 1). Is 211 divisible by 11 or by 13? No, as we can check quickly by long division.

And we only need to check possible prime factors up to the square root of 211 (which is between 14, the square root of 196, and 15, the square root of 225). If there is a pair of factors besides 211 and 1, one of the factors in the pair must be lower than the square root of 211 (and the other factor in the pair would be larger). Since there are no prime factors of 211 less than 14, we know that 211 is prime, and the second smallest prime factor of $$p(12) + 11 = 11 \times 211$$ is 211.

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Joined: 15 Jul 2014
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24 Nov 2014, 18:20
Can someone please explain what does the question mean by second smallest prime factor and why 11 is not the right answer?
Math Expert
Joined: 02 Sep 2009
Posts: 51158

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25 Nov 2014, 04:02
1
kritiu wrote:
Can someone please explain what does the question mean by second smallest prime factor and why 11 is not the right answer?

Because 11 is the smallest prime. The second smallest prime is 211.

If n is a positive integer greater than 1, then p(n) represents the product of all the prime numbers less than or equal to n. The second smallest prime factor of p(12)+11 is

A. 2
B. 11
C. 13
D. 17
E. 211

$$p(12)+11=2*3*5*7*11+11=11(2*3*5*7+1)=11*211$$. Both 11 and 211 are primes: 11 is the smallest prime of p(12)+11 and 211 is the second smallest prime of p(12)+11.

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Joined: 15 Jul 2014
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25 Nov 2014, 10:13
thanks Banuell. That helps!
Intern
Joined: 17 Jul 2016
Posts: 10
Location: India

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06 Feb 2017, 02:27
Hi

Why are we not considering 1 as one of the factors, then surely 11 would be the 2nd smallest?
Please correct me if i'm wrong.

Thanks
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Regards,
Lumin

Math Expert
Joined: 02 Sep 2009
Posts: 51158

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06 Feb 2017, 06:37
Ashandilya wrote:
Hi

Why are we not considering 1 as one of the factors, then surely 11 would be the 2nd smallest?
Please correct me if i'm wrong.

Thanks

Because 1 is not a prime number.
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Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1323
Location: Malaysia

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13 Feb 2017, 19:23
Bunuel wrote:
Official Solution:

If $$n$$ is a positive integer greater than 1, then $$p(n)$$ represents the product of all the prime numbers less than or equal to $$n$$. The second smallest prime factor of $$p(12) + 11$$ is

A. 2
B. 11
C. 13
D. 17
E. 211

The quantity $$p(12)$$ equals the product of all the primes less than or equal to 12. Thus, the number we are looking for is this:
$$2 \times 3 \times 5 \times 7 \times 11 + 11$$

$$= 11 \times (2 \times 3 \times 5 \times 7+1)$$ [factor out the 11]

$$= 11 \times (210 + 1)$$ [do the arithmetic]
$$= 11 \times 211$$
$$= \text{some large number}$$

We want to keep this number factored, and in fact we need to find its prime factorization. So, we ask: is 211 prime? Well, 211 cannot be divided evenly by 2, 3, 5, or 7, because 211 equals a multiple of all those numbers (210), plus 1. The "plus 1" means that 211 won't be divisible by any of the same factors as 210 (except for 1). Is 211 divisible by 11 or by 13? No, as we can check quickly by long division.

And we only need to check possible prime factors up to the square root of 211 (which is between 14, the square root of 196, and 15, the square root of 225).If there is a pair of factors besides 211 and 1, one of the factors in the pair must be lower than the square root of 211 (and the other factor in the pair would be larger). Since there are no prime factors of 211 less than 14, we know that 211 is prime, and the second smallest prime factor of $$p(12) + 11 = 11 \times 211$$ is 211.

Dear Bunuel, Do you have any example to illustrated the yellow highlight because I have no idea about what the statement is trying to convey.
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Joined: 11 Sep 2015
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02 May 2018, 08:23
Top Contributor
Bunuel wrote:
If $$n$$ is a positive integer greater than 1, then $$p(n)$$ represents the product of all the prime numbers less than or equal to $$n$$. The second smallest prime factor of $$p(12) + 11$$ is

A. 2
B. 11
C. 13
D. 17
E. 211

By the definition, p(12) = (11)(7)(5)(3)(2)

So, p(12) + 11 = (11)(7)(5)(3)(2) + 11
= 11[(7)(5)(3)(2) + 1] I factored out the 11
= 11[210 + 1]
= 11(211)

NOTE: 11 is prime and 211 is prime

The second smallest prime factor of p(12)+11 is . . .
We know that p(12)+11 has only 2 prime factors: 11 and 211
So, the smallest is 11, and the second smallest is 211

Cheers,
Brent
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Intern
Joined: 31 Jul 2018
Posts: 1

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08 Sep 2018, 10:33
I think this is a poor-quality question and I don't agree with the explanation. second smallest prime factor of this is 11
as it has 4 factors 1,11,211,2321 and 2 prime factors 11,211
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Joined: 11 Sep 2015
Posts: 3235

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08 Sep 2018, 13:39
Top Contributor
Aayush6970 wrote:
I think this is a poor-quality question and I don't agree with the explanation. second smallest prime factor of this is 11
as it has 4 factors 1,11,211,2321 and 2 prime factors 11,211

Keep in mind that 1 is not prime.

So, 11 is the smallest PRIME factor, which makes 211 the second smallest prime factor.
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Re: S97-10 &nbs [#permalink] 08 Sep 2018, 13:39
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# S97-10

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