BTGmoderatorDC 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

OA A

Source: Manhattan Prep

We have 3 Δ 47 = 3*5*7*9 ....*47. We see that the largest prime number that divides 3 Δ 47 is 47.

Note that (3 Δ 47) and (3 Δ 47) + 2 are consecutive odd numbers; thus, they are co-prime to each other. It means that no factor between them is common. Now since (3 Δ 47) + 2 is an odd number, and the largest prime number that divides (3 Δ 47) is 47, the smallest prime number that divides (3 Δ 47 + 2) must be greater than 47. Thus, the correct answer is A.

The correct answer: A

Hope this helps!

-Jay
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BTGmoderatorDC 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

OA A

Source: Manhattan Prep

Since none of the odd numbers between 3 and 47 (inclusive) divides into (3 �" 47) + 2, the smallest prime factor of (3 �" 47) + 2 must be greater than 47. The smallest prime number greater than 47 is 53, so y is at least 53.

Answer: A