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

BTGmoderatorLU wrote:Source: Manhattan Prep

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?

$$A.\, y>50$$ $$B.\, 30\le y\le50$$ $$C.\,10\le y<30$$ $$D.\, 3\le y<10$$ $$E.\, y=2$$

The OA is A

Given that a Δ b represents the product of all odd integers between a and b, inclusive, 3 Δ 47 = 3*5*7*...*47 = Odd number; thus, 3 Δ 47 = Odd + 2 = Odd.

Note that 3 Δ 47 and 3 Δ 47 + 2 are consecutive odd integers, thus, they cannot share any common factor except 1. For example. 33 and 35 do not share a common factor except 1.

Since 3 Δ 47 + 2 is an odd integer, it cannot have a prime factor 2 (even). So Option E is out.

Again, 3 Δ 47 = 3*5*7*...*47; thus every odd integer 3 through 47 is a factor of 3 Δ 47 = 3*5*7*...*47. This implies that none of the integers 3 through 47 is a factor of 3 Δ 47 + 2. Thus, the smallest factor for 3 Δ 47 + 2 would be greater than 47. Note that after 47, the next prime factor is 53, thus, y ≥ 53. Option A, y > 50 is correct.

The correct answer: A

Hope this helps!

-Jay
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BTGmoderatorLU wrote:Source: Manhattan Prep

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?

$$A.\, y>50$$ $$B.\, 30\le y\le50$$ $$C.\,10\le y<30$$ $$D.\, 3\le y<10$$ $$E.\, y=2$$

The OA is A

Since none of the odd numbers between 3 and 47 (inclusive) divide 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

BTGmoderatorLU wrote:Source: Manhattan Prep

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?

$$A.\, y>50$$ $$B.\, 30\le y\le50$$ $$C.\,10\le y<30$$ $$D.\, 3\le y<10$$ $$E.\, y=2$$

The OA is A

The above question is a nice twist on this similar official question: https://www.beatthegmat.com/number-prop ... 92872.html
Give it a try!

Cheers,
Brent