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If a and b are odd integers, a Δ b represents the product of

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If a and b are odd integers, a Δ b represents the product of [#permalink] New post 26 Dec 2012, 01:29
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If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2
[Reveal] Spoiler: OA

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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 26 Dec 2012, 02:00
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daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


(3 Δ 47) + 2 = 3*5*7*...*47+2 = odd + 2 = odd.

Now, 3*5*7*...*47 and 3*5*7*...*47 +2 are consecutive odd numbers. Consecutive odd numbers are co-prime, which means that they do not share any common factor but 1. For example, 25 and 27 are consecutive odd numbers and they do not share any common factor but 1.

Naturally every odd prime between 3 and 47, inclusive is a factor of 3*5*7*...*47, thus none of them is a factor of 3*5*7*...*47 +2. Since 3*5*7*...*47+2 = odd, then 2 is also not a factor of it, which means that the smallest prime factor of 3*5*7*...*47 +2 is greater than 50.

Answer: A.

Similar question from GMAT Prep: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 26 Dec 2012, 02:32
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daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2

Since each prime number from 3 upto 47 is a factor of (3 Δ 47) , none of them can be a factor of (3 Δ 47) + 2 . Also 48, 49 and 50 are not prime factors. And y cannot be 2 because (3 Δ 47) +2 is odd. Therefore y>50.
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 26 Dec 2012, 18:55
Bunuel wrote:
daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


(3 Δ 47) + 2 = 3*5*7*...*47+2 = odd + 2 = odd.

Now, 3*5*7*...*47 and 3*5*7*...*47 +2 are consecutive odd numbers. Consecutive odd numbers are co-prime, which means that they do not share any common factor but 1. For example, 25 and 27 are consecutive odd numbers and they do not share any common factor but 1.

Naturally every odd prime between 3 and 47, inclusive is a factor of 3*5*7*...*47, thus none of them is a factor of 3*5*7*...*47 +2. Since 3*5*7*...*47+2 = odd, then 2 is also not a factor of it, which means that the smallest prime factor of 3*5*7*...*47 +2 is greater than 50.

Answer: A.

Similar question from GMAT Prep: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Hope it helps.



Hi Bunuel,

Thanks for the solution above. Is their a significance of the term " Every Odd prime between 3 & 47"..It can very well be every prime between 3 and 47.
Please confirm.

Thanks
Mridul
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 27 Dec 2012, 01:18
Expert's post
mridulparashar1 wrote:
Bunuel wrote:
daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


(3 Δ 47) + 2 = 3*5*7*...*47+2 = odd + 2 = odd.

Now, 3*5*7*...*47 and 3*5*7*...*47 +2 are consecutive odd numbers. Consecutive odd numbers are co-prime, which means that they do not share any common factor but 1. For example, 25 and 27 are consecutive odd numbers and they do not share any common factor but 1.

Naturally every odd prime between 3 and 47, inclusive is a factor of 3*5*7*...*47, thus none of them is a factor of 3*5*7*...*47 +2. Since 3*5*7*...*47+2 = odd, then 2 is also not a factor of it, which means that the smallest prime factor of 3*5*7*...*47 +2 is greater than 50.

Answer: A.

Similar question from GMAT Prep: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Hope it helps.



Hi Bunuel,

Thanks for the solution above. Is their a significance of the term " Every Odd prime between 3 & 47"..It can very well be every prime between 3 and 47.
Please confirm.

Thanks
Mridul


Yes, that's correct.
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 30 Dec 2012, 07:32
Bunuel wrote:
daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


(3 Δ 47) + 2 = 3*5*7*...*47+2 = odd + 2 = odd.

Now, 3*5*7*...*47 and 3*5*7*...*47 +2 are consecutive odd numbers. Consecutive odd numbers are co-prime, which means that they do not share any common factor but 1. For example, 25 and 27 are consecutive odd numbers and they do not share any common factor but 1.

Naturally every odd prime between 3 and 47, inclusive is a factor of 3*5*7*...*47, thus none of them is a factor of 3*5*7*...*47 +2. Since 3*5*7*...*47+2 = odd, then 2 is also not a factor of it, which means that the smallest prime factor of 3*5*7*...*47 +2 is greater than 50.

Answer: A.

Similar question from GMAT Prep: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Hope it helps.


Hi Bunuel,

I just wanted to undertsand in what case 2 can be a smallest prime factor. For Eg if the Q. said that the smallest prime in (3 Δ 47) + 1.Then, the no (3 Δ 47) + 1 will be odd+1=even. Can we say 2 will be the smallest prime in this case.

Also, 2 consecutive integers will also be co-prime and therefore none of the factors in (3 Δ 47) will be factors of (3 Δ 47) + 1.

Thanks for your reply to my queries earlier.

Mridul
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 31 Dec 2012, 03:28
Expert's post
mridulparashar1 wrote:
Bunuel wrote:
daviesj wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 2, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


(3 Δ 47) + 2 = 3*5*7*...*47+2 = odd + 2 = odd.

Now, 3*5*7*...*47 and 3*5*7*...*47 +2 are consecutive odd numbers. Consecutive odd numbers are co-prime, which means that they do not share any common factor but 1. For example, 25 and 27 are consecutive odd numbers and they do not share any common factor but 1.

Naturally every odd prime between 3 and 47, inclusive is a factor of 3*5*7*...*47, thus none of them is a factor of 3*5*7*...*47 +2. Since 3*5*7*...*47+2 = odd, then 2 is also not a factor of it, which means that the smallest prime factor of 3*5*7*...*47 +2 is greater than 50.

Answer: A.

Similar question from GMAT Prep: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Hope it helps.


Hi Bunuel,

I just wanted to undertsand in what case 2 can be a smallest prime factor. For Eg if the Q. said that the smallest prime in (3 Δ 47) + 1.Then, the no (3 Δ 47) + 1 will be odd+1=even. Can we say 2 will be the smallest prime in this case.

Also, 2 consecutive integers will also be co-prime and therefore none of the factors in (3 Δ 47) will be factors of (3 Δ 47) + 1.

Thanks for your reply to my queries earlier.

Mridul


That is correct. The smallest prime of (3 Δ 47) + 1 is naturally 2, since (3 Δ 47) + 1 = even, and the smallest prime of any positive even integer is 2 (notice that 2 is the smallest prime).

Similar question to practice: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Hope it helps.
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 03 Jan 2013, 21:32
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(3*5*7*...*47)/y + 2/y = Integer

2/y is a fraction hence (3*5*7*...*47)/y also has to be a fraction. Only possibility is y>50
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If a and b are odd integers, a Δ b represents the product of [#permalink] New post 23 Mar 2013, 22:15
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 1, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2
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Last edited by Bunuel on 24 Mar 2013, 01:49, edited 2 times in total.
Edited the question.
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 23 Mar 2013, 22:29
Kindly check the question.
3 Δ 47 is odd ( product of all odd integers between 3 and 47 ).
3 Δ 47 + 1 is even, and hence the smallest prime factor is 2.
Answer should be E.

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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 23 Mar 2013, 22:49
HEY,
(3 Δ 47) AND (3 Δ 47) +1 should be consecutive numbers right? so they dont share the any common factors other than 1 .Hence the smallest prime factor should be more than 47 and hence E.
Please can some one explain whether my understanding is right or wrong ? if wrong please say where i made the mistake?
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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 23 Mar 2013, 23:01
Since (3 Δ 47) is odd and so (3 Δ 47) +1 is even. All even numbers have smallest factor as 2.
Hence E is the answer.

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Re: If a and b are odd integers, a Δ b represents the product of [#permalink] New post 24 Mar 2013, 01:50
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skamal7 wrote:
If a and b are odd integers, a Δ b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3 Δ 47) + 1, which of the following must be true?

(A) y > 50
(B) 30 ≤ y ≤ 50
(C) 10 ≤ y < 30
(D) 3 ≤ y < 10
(E) y = 2


Merging similar topics. Please refer to the solutions above.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If a and b are odd integers, a Δ b represents the product of   [#permalink] 24 Mar 2013, 01:50
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