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If a and b are odd integers, a Δ b represents the product of [#permalink]
26 Dec 2012, 01:29

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Question Stats:

50% (02:39) correct
50% (01:37) wrong based on 213 sessions

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

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
26 Dec 2012, 02:00

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Expert's post

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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.

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
26 Dec 2012, 02:32

2

This post received KUDOS

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. _________________

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
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.

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 _________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
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.

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.

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
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.

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 _________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
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.

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).

Re: If a and b are odd integers, a Δ b represents the product of [#permalink]
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]
10 Sep 2014, 07:30

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